{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# EART 70013 \n", " \n", "# Geophysical Inversion \n", " \n", "## Lecture 2 - Homework Solutions " ] }, { "cell_type": "markdown", "metadata": { "toc": true }, "source": [ "

Table of Contents

\n", "
" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import scipy.linalg as sl\n", "from pprint import pprint\n", "import scipy.interpolate as si\n", "from mpl_toolkits.mplot3d import Axes3D" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Homework" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Homework - Optimisation - simple example\n", "\n", "Consider the problem from the lecture\n", "$$ \n", "f(\\boldsymbol{x}) = \n", "1+2x + 4y + x^2+2xy+3y^2\n", "$$\n", "\n", "Compute the gradient vector, and by setting it equal to zero and writing as a matrix equation,\n", "solve for the stationary point. \n", "\n", "Plot the function via a contour plot in 2D, and add the stationay point you've computed to verify it is indeed a minima (refer to the image from the lecture)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution \n", "\n", "The gradient vector is\n", "\n", "$$\\nabla f= \n", "\\begin{pmatrix}\n", "2 + 2x + 2y\\\\\n", "4 + 2x + 6y\n", "\\end{pmatrix}\n", "$$\n", "\n", "We can write this as a linear system:\n", "\n", "$$\n", "\\nabla f = \\begin{pmatrix}\n", "2 & 2\\\\\n", "2 & 6\n", "\\end{pmatrix}\n", "\\begin{pmatrix}\n", "x\\\\\n", "y\n", "\\end{pmatrix}\n", "+\n", "\\begin{pmatrix}\n", "2\\\\\n", "4\n", "\\end{pmatrix}\n", "$$\n", "\n", "So $\\nabla f = 0$ when \n", "\n", "$$\n", "\\begin{pmatrix}\n", "2 & 2\\\\\n", "2 & 6\n", "\\end{pmatrix}\n", "\\begin{pmatrix}\n", "x\\\\\n", "y\n", "\\end{pmatrix}\n", "=\n", "\\begin{pmatrix}\n", "-2\\\\\n", "-4\n", "\\end{pmatrix}\n", "$$\n", "\n", "Let's use a contour plot to visualise the function in 2D, solve for the minimum (the stationary point) and plot it:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "dx = 0.1\n", "x = np.arange(-2.0, 2.0, dx)\n", "y = np.arange(-2.0, 2.0, dx)\n", "X, Y = np.meshgrid(x, y)\n", "f = 1. + 2.*X + 4.*Y + X**2 + 2.*X*Y + 3.*Y**2 \n", "\n", "fig = plt.figure(figsize=(5, 5))\n", "ax1 = fig.add_subplot(111)\n", "\n", "cs = ax1.contour(X, Y, f, 10)\n", "ax1.clabel(cs, inline=1, fontsize=10)\n", "ax1.set_title('Contour plot of function and the minima location')\n", "ax1.set_xlabel('$x$', fontsize=12)\n", "ax1.set_ylabel('$y$', fontsize=12)\n", "\n", "# solve the linear system for x,y\n", "A = np.array([[2,2],[2,6]])\n", "b = np.array([-2,-4])\n", "x = sl.solve(A,b)\n", "ax1.plot(x[0],x[1],'r*')\n", "#plt.savefig('simple_optimisation.png')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Homework - $2\\times 2$ linear systems\n", "\n", "Recall from the lecture the system\n", "\n", "\\begin{eqnarray*}\n", " 2x + 3y &=& 7 \\\\[5pt]\n", " x - 4y &=& 3,\n", "\\end{eqnarray*} \n", "\n", "and the following plot which demonstrated a unique solution." ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# unique solution\n", "x = np.linspace(-1,5,100)\n", "y1 = -(2./3.)*x + (7./3.)\n", "y2 = (1./4.)*x - (3./4.)\n", "\n", "fig = plt.figure(figsize=(5, 5))\n", "\n", "ax1 = fig.add_subplot(111)\n", "\n", "ax1.set_xlabel(\"$x$\", fontsize=14)\n", "ax1.set_ylabel(\"$y$\", fontsize=14)\n", "ax1.set_title('Two lines', fontsize=14)\n", "ax1.grid(True)\n", "\n", "ax1.plot(x,y1,'b', label='$2x+3y=7$')\n", "ax1.plot(x,y2,'r', label='$x-4y=3$')\n", "\n", "ax1.legend(loc='best', fontsize=14);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "You were asked in class what other situations are possible and to construct examples and the corresponding plots - try to come up with and plot these situations.\n", "\n", "Hint: I used the following example as a starting point which was also discussed in the lecture\n", "\n", "$$\n", "\\left(\n", " \\begin{array}{rr}\n", " 2 & 3 \\\\\n", " 4 & 6 \n", " \\end{array}\n", "\\right)\\left(\n", " \\begin{array}{c}\n", " x \\\\\n", " y\n", " \\end{array}\n", "\\right) = \\left(\n", " \\begin{array}{c}\n", " 4 \\\\\n", " 8\n", " \\end{array}\n", "\\right),\n", "$$\n", "\n", "with a second RHS vector of \n", "\n", "$$\n", "\\left(\n", " \\begin{array}{c}\n", " 4 \\\\\n", " 7\n", " \\end{array}\n", "\\right)\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the problem\n", "\n", "$$\n", "\\left(\n", " \\begin{array}{rr}\n", " 2 & 3 \\\\\n", " 4 & 6 \n", " \\end{array}\n", "\\right)\\left(\n", " \\begin{array}{c}\n", " x \\\\\n", " y\n", " \\end{array}\n", "\\right) = \\left(\n", " \\begin{array}{c}\n", " 4 \\\\\n", " 8\n", " \\end{array}\n", "\\right),\n", "$$\n", "\n", "where the matrix on the LHS clearly has a zero determinant.\n", "\n", "Considering now the values in the RHS vector as well, the second equation is simply twice the first, and hence a solution to the first equation is also automatically a solution to the second equation. \n", "\n", "If we think geometrically, and interpret the two equations as constraints, they are **both** constraining our $x,y$ values to the same 1D subspace - any solution along this line in 2D satisfies **both** equations, and hence any of them is a solution to our linear system.\n", "\n", "We hence only have one *linearly-independent* equation here, and our problem is under-constrained: we effectively only have one equation for two unknowns and this problem has *infinitely many* possibly solutions (or said another way, we have *existence*, but *non-uniqueness*): e.g. $\\boldsymbol{x}=(2,0)^T$ is a solution, so is $\\boldsymbol{x}=(-1,2)^T$, etc.\n" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# consider the following situation\n", "x = np.linspace(-1,5,100)\n", "y1 = -(2./3.)*x + (4./3.)\n", "y2 = -(4./6.)*x + (8./6.)\n", "\n", "fig = plt.figure(figsize=(5, 5))\n", "\n", "ax1 = fig.add_subplot(111)\n", "\n", "ax1.set_xlabel(\"$x$\", fontsize=14)\n", "ax1.set_ylabel(\"$y$\", fontsize=14)\n", "ax1.set_title('Two lines', fontsize=14)\n", "ax1.grid(True)\n", "\n", "ax1.plot(x,y1,'b', label='$2x+3y=4$')\n", "ax1.plot(x,y2,'r', label='$4x+6y=8$')\n", "\n", "ax1.legend(loc='best', fontsize=14);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following are all true, but not all equivalent statements:\n", "\n", "1. The lines now intersect at infinitely many points - not at all points in 2D, but along a 1D subspace embedded within 2D.\n", "\n", "2. We therefore have a continuous family of $(x,y)$ pairs of values that satisfy both equations simultaneously.\n", "\n", "3. The linear/matrix system has infinitely many solutions (our previous *no solution existence* situation has become a *non-uniqueness* situation).\n", "\n", "4. The corresponding matrix has zero determinant.\n", "\n", "5. The matrix has no inverse.\n", "\n", "Let's check point 4:\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "The determinant is: -6.661338147750939e-16\n", "This is effectively zero: True\n" ] } ], "source": [ "A = np.array([[2,3],[4,6]])\n", "print('The determinant is: ',sl.det(A))\n", "print('This is effectively zero:',np.allclose(0,sl.det(A)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Any pair of values that satisfies \n", "\n", "$$x = (4-3y)/2$$\n", "\n", "is an exact solution. Typically we will write solutions of this type in the form \n", "\n", "\\begin{align*}\n", "x &= (4-3\\alpha)/2\\\\\n", "y &= \\alpha\n", "\\end{align*}\n", "\n", "where $\\alpha$ is an arbitrary constant." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we instead replaced the RHS vector with $(4,7)^T$, then the two equations would now be contradictory: in this case we have *no solutions* (or *non-existence*)." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# consider the following situation\n", "x = np.linspace(-1,5,100)\n", "y1 = -(2./3.)*x + (4./3.)\n", "y2 = -(4./6.)*x + (7./6.)\n", "\n", "fig = plt.figure(figsize=(5, 5))\n", "\n", "ax1 = fig.add_subplot(111)\n", "\n", "ax1.set_xlabel(\"$x$\", fontsize=14)\n", "ax1.set_ylabel(\"$y$\", fontsize=14)\n", "ax1.set_title('Two lines', fontsize=14)\n", "ax1.grid(True)\n", "\n", "ax1.plot(x,y1,'b', label='$2x+3y=4$')\n", "ax1.plot(x,y2,'r', label='$4x+6y=7$')\n", "\n", "ax1.legend(loc='best', fontsize=14);" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For this case some of our previous statements will still hold (as the LHS matrix is unchanged), but some will change (as the RHS vector is different):\n", "\n", "\n", "1. The lines do not intersect at any point.\n", "\n", "2. No single $(x,y)$ pair of values can satisfy both equations simultaneously.\n", "\n", "3. The linear/matrix system has no solution (we are in a *no solution existence* situation).\n", "\n", "4. The corresponding matrix has zero determinant.\n", "\n", "5. The matrix has no inverse.\n", "\n", "\n", "So 4 and 5 haven't changed, but the change in RHS vector has changed 1,2,3 dramatically.\n", "\n", "\n", "To summarise, other than cases with a unique solution, 2D linear systems can also have no solutions or infinitely many solutions. If we want existence **AND** uniqueness of solution of course both scenarios aren't good!" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Homework - Using row operations to solve the linear system and find the inverse matrix\n", "\n", "Consider the linear system\n", "\n", "$$\n", " \\begin{pmatrix}\n", " 2 & 3 \\\\\n", " 1 & -4 \n", " \\end{pmatrix}\n", " \\begin{pmatrix}\n", " x \\\\\n", " y \n", " \\end{pmatrix}=\n", " \\begin{pmatrix}\n", " 7 \\\\\n", " 3 \n", " \\end{pmatrix} \n", "$$\n", "\n", " \n", "form the augmented matrix $[A|\\boldsymbol{b}|I]$ and perform row operations to obtain $[I|\\boldsymbol{x}|A^{-1}]$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution\n", "\n", "We can perform these operations in a way that is easy to keep track of by forming the augmented matrix (we add the RHS vector as an additional column). \n", "\n", "In this demonstration where we also want to form the inverse matrix let's additionally add columns corresponding to the inverse matrix. \n", "\n", "The notation above the arrows should hopefully be clear, it's encoding the operations we are performing on the rows, with '(1)' and '(2)' identifying the first and second row, respectively. '$(1)\\leftarrow (1)-2(2)$' means replace the first row with the first row minus twice the second row - the operation we performed above.\n", "\n", "When we perform these operations on the rows of the matrix $A$, we also apply them to the additional columns in our augmented matrix:\n", "\n", "\\begin{align*}\n", "[A \\, | \\, \\boldsymbol{b} \\, | \\, I] = \n", "&\\left[\n", " \\begin{array}{cc|c|cc}\n", " 2 & 3 & 7 & 1 & 0 \\\\\n", " 1 & -4 & 3 & 0 & 1 \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(1)\\leftarrow (1)-2(2)}\n", "\\left[\n", " \\begin{array}{cc|c|cc}\n", " 0 & 11 & 1 & 1 & -2 \\\\\n", " 1 & -4 & 3 & 0 & 1 \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(1)\\leftarrow (1)/11}\n", "\\left[\n", " \\begin{array}{cc|c|cc}\n", " 0 & 1 & 1/11 & 1/11 & -2/11 \\\\\n", " 1 & -4 & 3 & 0 & 1 \n", " \\end{array}\n", "\\right]\\\\\n", "&\\xrightarrow{(2)\\leftarrow (2)+4(1)}\n", "\\left[\n", " \\begin{array}{cc|c|cc}\n", " 0 & 1 & 1/11 & 1/11 & -2/11 \\\\\n", " 1 & 0 & 3+4/11 & 4/11 & 1-8/11 \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(1)\\leftrightarrow (2)}\n", "\\left[\n", " \\begin{array}{cc|c|cc}\n", " 1 & 0 & 3+4/11 & 4/11 & 1-8/11 \\\\\n", " 0 & 1 & 1/11 & 1/11 & -2/11 \n", " \\end{array}\n", "\\right]\n", "\\end{align*}\n", "\n", "Now the fact that we have transformed $A$ into $I$ via these so-called *row operations* means that what was originally in the $\\boldsymbol{b}$ position is now the solution $\\boldsymbol{x}$, and what was originally $I$ is now $A^{-1}$:\n", "\n", "$$\\boldsymbol{x} =\n", "\\begin{pmatrix}\n", "3+4/11\\\\\n", "1/11\n", "\\end{pmatrix}=\n", "\\begin{pmatrix}\n", "37/11\\\\\n", "1/11\n", "\\end{pmatrix},\\qquad\n", "A^{-1} = \n", "\\begin{pmatrix}\n", " 4/11 & 1-8/11 \\\\\n", " 1/11 & -2/11 \n", "\\end{pmatrix}= \n", "\\begin{pmatrix}\n", " 4/11 & 3/11 \\\\\n", " 1/11 & -2/11 \n", "\\end{pmatrix}= \\frac{-1}{11}\n", "\\begin{pmatrix}\n", " -4 & -3 \\\\\n", " -1 & 2 \n", "\\end{pmatrix}\n", "$$\n", "\n", "\n", "We can check this answer against our expression for the inverse of a $2\\times 2$ linear system.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Homework - Formula for the inverse of a $2\\times 2$ matrix using row operations\n", "\n", "Starting from the augmented matrix\n", "\n", "$$\n", "[A \\, | \\, I] = \n", "\\left[\n", " \\begin{array}{rr|rr}\n", " a & b & 1 & 0 \\\\\n", " c & d & 0 & 1 \n", " \\end{array}\n", "\\right]\n", "$$\n", "\n", "use row operations to turn the matrix on the left into the identity matrix. Confirm that the matrix you obtain on the right is equivalent to the general formula for the inverse matrix.\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solution \n", "\n", "\\begin{align*}\n", "[A \\, | \\, I] = \n", "\\left[\n", " \\begin{array}{cc|cc}\n", " a & b & 1 & 0 \\\\\n", " c & d & 0 & 1 \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(1)\\leftarrow c(1);\\, (2)\\leftarrow a(2)}\n", "\\left[\n", " \\begin{array}{cc|cc}\n", " ca & cb & c & 0 \\\\\n", " ac & ad & 0 & a \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(2)\\leftarrow (2)-(1)}\n", "\\left[\n", " \\begin{array}{cc|cc}\n", " ca & cb & c & 0 \\\\\n", " 0 & ad - bc & -c & a \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(2)\\leftarrow (2)/(ad-bc)}\n", "\\left[\n", " \\begin{array}{cc|cc}\n", " ca & cb & c & 0 \\\\\n", " 0 & 1 & -c(ad-bc) & a/(ad-bc) \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(1)\\leftarrow (1)-cb(2)}\n", "\\left[\n", " \\begin{array}{cc|cc}\n", " ca & 0 & c + c^2b/(ad-bc) & -acb/(ad-bc) \\\\\n", " 0 & 1 & -c/(ad-bc) & a/(ad-bc) \n", " \\end{array}\n", "\\right]\n", "\\xrightarrow{(1)\\leftarrow (1)/ca}\n", "\\left[\n", " \\begin{array}{cc|cc}\n", " 1 & 0 & 1/a + cb/(a(ad-bc)) & -b/(ad-bc) \\\\\n", " 0 & 1 & -c/(ad-bc) & a/(ad-bc) \n", " \\end{array}\n", "\\right]\n", "\\end{align*}\n", "\n", "and finally note that \n", "\n", "$$ \\frac{1}{a} + \\frac{cb}{a(ad-bc)} = \\frac{ad-bc + cb}{a(ad-bc)} = \\frac{ad}{a(ad-bc)} = \\frac{d}{ad-bc}$$\n", "\n", "and so we are left with\n", "\n", "$$A^{-1} =\n", "\\begin{pmatrix}\n", "1/a + cb/(a(ad-bc)) & -b/(ad-bc) \\\\\n", "-c/(ad-bc) & a/(ad-bc) \n", "\\end{pmatrix} = \n", "\\begin{pmatrix}\n", "d/(ad-bc) & -b/(ad-bc) \\\\\n", "-c/(ad-bc) & a/(ad-bc) \n", "\\end{pmatrix}\n", "= \n", "\\frac{1}{ad-bc}\n", "\\begin{pmatrix}\n", "d & -b \\\\\n", "-c & a \n", "\\end{pmatrix}\n", "$$\n", "\n", "as required.\n", "\n", "Note of course that an early step in the derivation above involved dividing through by $ad-bc$, and therefore required that this quantity be non-zero. Of course we recognise this as the determinant of the matrix!" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Homework - Calculating the rank and the null space\n", "\n", "How can we compute the rank and the null-space for a given matrix?\n", "\n", "We can do this using row operations. \n", "\n", "We already stated that row operations, when performed on the augmented matrix, results in an updated linear system with the same solution. We did this since the updated augmented system is constructed such that it is trivial to solve. \n", "\n", "It's also the case that performing row operations (on a matrix itself or the augmented system) does not change the rank of a matrix (or the augmented system).\n", "\n", "Hence we can compute the rank of a matrix by performing row operations on it to transform it to the simpler (reduced) row echelon form.\n", "\n", "We can then easily read off the rank of the transformed matrix as the number of linearly independent rows (equivalently the number of linearly independent columns), and this is also the rank of the matrix in its original form.\n", "\n", "\n", "### A worked example - please read through \n", "\n", "\n", "For example consider the following square matrix case\n", "\n", "$$ A =\n", "\\begin{pmatrix}\n", "-2 & -4 & -20 \\\\\n", "2 & 6 & 24 \\\\\n", "2 & 10 & 32\n", "\\end{pmatrix}\n", "$$\n", "\n", "Let's multiply the first row by -1/2 in order to generate a one in the first entry of the first row (we could choose to swap rows first, so of course there is no unique way to go about this process), this gives us\n", "\n", "$$\n", "\\begin{pmatrix}\n", "1 & 2 & 10 \\\\\n", "2 & 6 & 24 \\\\\n", "2 & 10 & 32\n", "\\end{pmatrix}\n", "$$\n", "\n", "Remove the entries below that \"1\" that's now in the top left by subtracting multiples of the first row from the second and third:\n", "\n", "$$\n", "\\begin{pmatrix}\n", "1 & 2 & 10 \\\\\n", "0 & 2 & 4 \\\\\n", "0 & 6 & 12\n", "\\end{pmatrix}\n", "$$\n", "\n", "Now turn the first entry in the second row into a \"1\"\n", "\n", "$$\n", "\\begin{pmatrix}\n", "1 & 2 & 10 \\\\\n", "0 & 1 & 2 \\\\\n", "0 & 6 & 12\n", "\\end{pmatrix}\n", "$$\n", "\n", "and remove the value below, as well as the value above, by subtracting the appropriate multiples of the second row from the first and the third:\n", "\n", "$$\n", "\\begin{pmatrix}\n", "1 & 0 & 6 \\\\\n", "0 & 1 & 2 \\\\\n", "0 & 0 & 0\n", "\\end{pmatrix}\n", "$$\n", "\n", "You can hopefully see that this has two linearly independent columns (we can easily see how the first two columns can be combined to form the third), \n", "\n", "it also has two linearly independent rows (as we expect). \n", "\n", "The rank of the matrix in this final form is thus 2, and therefore $\\text{rank}(A)=2$.\n", "\n", "This means that the dimension of the range of $A$ is two, i.e. the range will span a plane within $\\mathbb{R}^m = \\mathbb{R}^3$.\n", "\n", "\n", "We can check this using `numpy`:" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "array([[ -2, -4, -20],\n", " [ 2, 6, 24],\n", " [ 2, 10, 32]])\n", "2\n", "array([[1, 0, 6],\n", " [0, 1, 2],\n", " [0, 0, 0]])\n", "2\n" ] } ], "source": [ "# compute the rank of the matrix in original form\n", "A = np.array([[-2, -4, -20], [2 , 6 , 24], [2, 10, 32]])\n", "pprint(A)\n", "print(np.linalg.matrix_rank(A))\n", "\n", "# compute the rank of the matrix in a form after we've performed row operations\n", "Arref = np.array([[1, 0, 6], [0 , 1 , 2], [0, 0, 0]])\n", "pprint(Arref)\n", "print(np.linalg.matrix_rank(Arref))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Let's create a series of points in 3D space (equivalently vectors extending from the origin) and see how they transform under multiplication of $A$. If above is all correct the dimension of the mapped points/vectors whould be 2 - it should be a 2D plane in 3D." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "data": { "image/png": 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Egv2ABQaYcMxW78icMFi9YyQSQUVFBaqrq2EwGDAwMLAva5eSsJycnEQoFMLCwgIaGhpgMplEKvwwQ8gdoPSVQjYVYrLEpH9R1tbWMDs7i+PHj3OPRQCgFy4g+vTT0LzwAqR7781ZMylJEqamprCzs4P+/n4YDIaMj1X6WS3pM7gvX76MlpYWuN1uTExMIBwOc7PxmpqaXcdxWCkbHITtP/5HIBKBQ6eD51//FZt1dTwaazab+TmVl5cDSLzm6eKSpcSFuBQIBKVCqd5RkiSsrq6CEAKHw8HFYzQa5cLR4/FAkiRUVVXBYrHAbren3PCvrKwcyDrFhGVNTQ3m5+cBAIODgwDAayyFsDyU3AdAiMmDRu4dGY/HMTExgVgshoGBAej1+l2Ppxcu7Epty9FoNJAkCcFgEENDQ6irq8OZM2cUv2RK4rEU6WWlMYk+nw8ulwvLy8vcbNxqtXI/yMOI3ESdUIryK1egv/deHo1lqf75+Xn4/f5EKUJSXLJOe0mSEAqF+D6DwSC/+2cLoEAgEORCTb0ju2GNxWLY2NiAx+OB1+uFRqPhzTJtbW2K15bDhFarRVtbG9ra2hAKhbC+vp5TWK6urqKlpUUIy73klVeA558H7rsPuOMO9tvnC93d4bzy32IojUQcHh5Ga2srmpqaCv4iEEIQCoVw7do19PX1ZZ3RzZ6Dicq9+vKxhay6uhodHR0pZuM3b94EgBTLnv2yIcpFuol67O67+d8IISgrK0NZWRmamppSUv2zs7MIBAKoqKjg52U2mwEAU1NTOHbsGK8JYpFLVnMpFkCBQADkV+8YCAR41NHtdkOr1cJut6O+vh7d3d0HtqYWGpyQr4Mmk2mXsBwaGgKllDfvmEwm3Lx5E42NjSkRS/nrJCiSV14B3vY23gCMH/84ISgLTHEDQkwWDRv5NzQ0hOPHj2NpaQnLy8s4efIkr8UrBEmSMDMzg2AwiLuTQigbBzWbW8lsfHt7G06nk1v2SJIEo9EIi8VyYE0tchP14dpaHD1/PuNj01P9lFL4/X643W5MT08jFAqhoqKCm8Kzu2oWRWaLnRCXAsGbD3m9I/MAZqUzQGq9oyRJvFmGNcyUl5ejuroaLS0tqKqqAiEETU1NB3Ie6eS7hmW7BmUTluFwGJFIJCViyZ5fCMsS8PzzCSEZjyf+//zz8uhkQQgxWQTMO5JZIAwODsJoNGJgYKCou8dAIIDh4WFYrVZUVlaqqkvcqzR3vuh0OthsNthsNgAJK54bN27A5XJhdXUVBoOBd4rvlQ1RJpiJuvfKlby2I4SgoqICFRUVaGlpAaUUPp+Pi8tIJILKykqeFme2GEJcCgRvfOT1jizyyNbeUCiE+fl5nDhxgjfLMOHo8XgQjUZRWVmJ6upqdHd387nXDJ/Pl9cUtFJTijVKzT7SheXFixcVI5ZyYQkkrjdCWOZAKZ19332JiCSLTN53X9FPI8RkAaQXDXs8HgQCARw5coTb7RTK+vo6pqencfz4cZSVlWF7e1vVdkqRycOAwWBAWVkZbDYbt+Vxu91YXFzc1fhSVlZ2aI47G4QQVFVVwWAw4NSpUyni8saNG4hEIrxJyWKxZBSX8k5xIS4FglsDFnWUp6zlay8TNizyGA6HuScwAN4s09TUBKPReFCnsS8UEtAwGo28Cz0cDmdMhacLS9a8I4SljEzp7DvuSPx7d81kwQgxmSdy70gAmJ2dxdbWFsrKyooSkpIkYXx8HOFwmDfsxGKxoupU8t1Wc+kSdC+9hNjdd0PKkgYuBpPJhMbGRjQ2NqY0vszNzSEQCKC8vJyLS1abWEpKHa1lFw1WRwok3kuv18t9LmOxWIq4NBgMKQX4Xq8X8XgcdXV1PHIpFkSB4HAgSRKPPCrVO7K0NRuXy6KOrFkmFovBarWio6Pj0DYoHibkNf9GoxGtra1obW1VFJYNDQ0wm82glGJiYgLV1dWoq6sTwpKRLZ3NRCXwevTyD/9QWAPtB/IURjgcxsjICCwWC/r7+3Hx4sW89iX/wgQCAQwNDcFut+PYsWO7mmnUQAhBOBzGzZs3YTKZYLFY8kqPaC5dQtn73gdEIjAYDAg89dSeCUqGUuPLzs4O3G43JicnEQ6HU9LHpWAvm5MYGo0GFouFW0HF43EuLpeWlngHPBOXrPaSiUvg9dogIS4Fgv0j3d8xFAphY2OD+9XK6x1ZLTVLW/v9fr72NjY2oqenB9FoFOPj46itrS3oePar9l0NhaydpVxvlYTl8PAwF5axWCzFe5OJ/je1sFSTzpZHL4EfAygr5KmEmFRBelp7a2sLU1NT6O3tLWiRkHdcr66uYm5uDsePH+eRLfnj1ArCUCiEiYkJ9PT0QHflCvD88zC2t2NQ5ahEuXUOjUSge+klRPZYTKaTyYbI7XZjbGwMfr8fk5OTXIQVYomxH2IyHTY3nAli1gEffv55bL/0Elx9fYicPcunVOh0upTIJZB4beRp8TflwigQlJhs9Y5A4ru6ubmJpqYmnnFgkcdwOMybZdrb21FeXr7rO3mYxGC+UErhdDrh8Xh4ac9+nIvaKW7pwnJjYwMulws7Ozuw2+08YvmmFpZq0tny6CVQsM+UEJM5kHtHslB6IBBQNA5XK1Q0Gg2i0Simp6cRjUbR39+vKIzULESUUkxPT8Pr9aKnpwd109Mo++hHgUgEjQrm3CyNbLVaU9LIcuscGAwp1jnFUqiAk9sQtbe34/Lly6irq+OjHyml3K7HYrGoano6CDGZjlarhW1qCmWf+hQQiaBdr8fUF76A7cpKbvCbfl7shobZTwlxKRDkj1xcsABBpnrHWCzG6+GvXbuGeDyOyspKWCwWnl7Nxa0kJuVm6Nvb29weraamBjqdDgsLC9je3sbk5CQaGxtRWVmZc5/7Ec1kwjIQCMBisSASiWBkZATxeJzXWL5hhKWyN2R2lNLZmZpx4vFooYcmxGQG0r0jA4EARkZGYLfb0dvbm/EOVM2HUpIkvPrqq2hubkZLS0vGbXLtKxwOY2hoCDU1NbDb7QlhkcOcm1ncyNPIVqsVNadPA0nrnL2smSwGQkhKhI/ZELlcLszNzfH0MvO4VLIhOgxiEkiNBAOAbWQEhl/6JQC7z4sQssu7k1lSpYtLNqHnllogBYI9Qh7hl9c7Aq/XO7N1IhwOp0yW0Wg0qKyshFarxW233XYg0772UoxGIpGUznL5+tne3o7XXnsNR44cAZBopDSbzZiZmUFVVRWmp6cRDAZRX1+PxsbGFOujUlDI2kUphcFggN1uT4lY5hKWq6urcDgc0Ov1h3vdzNRMU+z28ujlH/7h2wo9PCEmFWDGsQsLC+jo6MDKygpu3rypmIpmsGk1uXwUV1ZWsLOzgzNnzhRcRwMATqcT4+PjOHr0KGw2G6amphLRqxzm3HKLm13TbDQaWN73PlitVlTHYoe+WDzdhigajcLtdmNjYwPT09PQ6/VcfFZWVvKF+aC8LuWkv0+hCxfALlVK57W9vY2trS3u3cmilkw0M3E5ODiIo0ePcmHJOsUP9SIpEJQAeb0jE49utxuRSAR1dXW76h2DwSAXUzs7O9Dr9bBYLLDZbOjq6oJWq0U8Hsf169cLFpKHKTIZDof5+Xq9Xmi12l3nmwuNRgO73Q673c4n84yPjyMSifDfp0dsS+lNmWs7+XPJU+GRSGSXsGxoaEBZWRkWFxfR0NDAn/fQRiyL9YZU04zz2c8K0/JSwbwjY7EYtra2EqnhoSHctbQEaDSKM7SB18VkJuLxOG7cuIF4PA6LxVLwnRylFLOzs3A6nTh79iyfw8oWLbk591BtLXqzRBiVptlsb2/z7mqNRsMXoKqqqkMhwrKh1+tRX1+P+vp6AInF0+12Y3l5GT6fDyaTCZWVldzK4yAXCvn7dLOjA4Zz5zI+Vq/Xo66uDnV1dQBejyhsbGxgamoqRTQHg0FeMhGJRPhccY1Gw8UlG9N2qBZKgSBPWL2jPGUtX4NZqjoUCoEQgp2dHR55DAQCKCsrQ3V1NZqbm1FRUZFxfStWDB7U9qFQCNFoFDdu3IDP5+NiudBJOulrpk6ng8PhgMPhSBFrkiTxKGAhx74Xa7PBYEBLSwtaWlr4sY6OjiIejyMYDCISiaCsrOzwpMKL8YbMlArfA29JOUJMJpGntQkhCAQC8Hg86Fxfh+P/+r8Sb8BjjyH69NOKgpJFh5TY2dnB8PAwmpub0dzcjOvXrxdkRBuJRDA8PIyKigqcO3cuZfGTp9njAwOQzp+H9/LlvPav1WpRW1vLI6aRSARXr17F2toaJicnYUw281itVsVi82wchHAxGo38bplFIjY2NhAMBnHlypWU+dvpZsH7ATNR35magi0PoW4wGBRF88rKCgKBAAYHB1Oarthnk4lLJiT1ej1PiwtxKTjsyJtl5PWObN2Tm1ezJrfNzU14vV5sbGzwkaidnZ2qPW2L/U7s1/ZKkVaj0QhJkuBwOFBZWbmnwQC5WAuHw1hbW8Pg4CBisRgMBgMiSZcQtRSa5lazXbqwfPnll7mwlEcs04Ulm5e+58KyGG/IbKnwPfCWlCPEJHZ7R968eROrq6swm81oeO21lC5nzQsvIK4gJjN1Xi8vL+PmzZs4ceIEL1guJPWxvb2N0dFRHDlyhIuIXBT7gTcYDNDr9ejt7QUA7gk5Pz8Pv9+P8vJyPs1mLzwhSwmzIaqvr8fOzg6OHz++a0Ri+hSbbJQydaWmPCIbctG8s7OD3t5ebkPELipyccmIRCJ8rjhbKOU1lwLBQZJPvSNLabP6P0mSUFVVBaPRCKvViqNHjxZ0DMWmqfcqzS2f4c3EI3ODkEdar1y5krE0a68wGo18ms3y8jKWl5dx7do16HQ6NDY2or6+PqsTR6nS3GowGAwwGAzo7+/nEcuxsTHEYjHU19fDbrdzYcl0ArDHwlKtN2S+26rZvgje9GJSPhIxGo1iZGQEZWVlOH/+PC5evAjp3nuhlXU5S/feq7if9DR3LBbD2NgYAGBgYCCl/jBbFDMdSinm5+exvr6OM2fOZBRtSotWqRcxs9kMs9mc0szjcrl4Mw8z5q6pqUm5Cy30OOQm6vGBgZKcA1twlOpHmcfljRs3EI1GU84nffErZSqmWDGZjtwYHnj9JoB19MunDsnLLdLFpSRJKC8vF+JSsC+kp6zj8TivC08fmZduDq7ValFdXQ2LxYK2tjb+fXW5XHC5XAUf00FH69m6Lm+elKfpa2pq0NbWlnemaL8wGAywWCzo6emB3+/H2toarl69CqPRiMbGRtTV1e2qzS90bS12TU6PWG5sbGBsbAzRaJSn7eXC8rXXXsPRo0dhMBgKF5alTmfvcSo7G29aMZnuHel2uzE+Pr4r8kcvXED06aeheeEFSPfeq6pm0ufzYXh4GK2trWhubt71WLX+kdFoFMFgEMFgEP39/Vkv6Ptd6C0XY8wTkhlzLy8vQ5Ik3hlYSEpfyUQdJRA0mRYcjUaDqqoqVFVVoa2tDZIkwePx8AifJEkpRuNsm1JQajGZTvpNQHqEWZ7uLytL+NVSSvHqq6/i7NmzAIA/+7M/w5/+6Z/yvwsExaCm3lGj0SAQCECr1cLv9/N6R7/fD6PRiOrqajQ0NGSt/zvoBphCn5/d3G5tbcHr9WJ9fb2gNH0pKdbmp7y8HF1dXejq6oLP58Pa2hpmZ2dRXl7OhSVbBw9CTMoxGAy8LC2TsPT7/SklREDimsDqLHMey16ks/c4lZ2NN6WYlKe1KaWYmZmB2+1OaWhJefyFC4qpbTlMTC4tLWFxcREnT55MSSnKUbPAeDwejI6OQq/X49ixYznPSWmf+7nYyKe+dHR0JArff/pTSF/+MmIdHZi4/XbU1dXBarWqqt9JN1HXvvgiSAnusvLxAk03GmfNScwLMhKJwO12o6qqKu9idjl7LSblKE0dYn5ys7OzCAQC/KIl73z/yU9+gv/+3/97zv1//OMfx/e+9z3U19djZGRE6fkJgL8G8ACAAICPUUqvJf/2ruTftAD+jlL6aKnOW3Cw5FPvyFwmtre3EQwGcfnyZZSXl8NisaC1tTXr8IV0DlpMAuoyM/Jz3t7eRjgcRkVFBbRaLaxWK44cOXKgkcdSv4ZsOEV3dze8Xi/W1tYwPT2NqqoqWK3WQ9O4A2QWlj6fDzdv3kRjY2NKxJJZtuUUlnuVzt7DVHY23lRiMt07MhQKYWRkBFarFf39/UV9ECmlmJqagslkwsDAQFZxka3zm1KKxcVFLC8v49SpUxgcHFT1/PuR5s4Hw6uvwvIf/gMQicCu02H9n/4JgbY2rK6uYmJigtcysVRr+mufbqIeveuukiwUhS446c1Jfr8fY2Nj2NzcxPT0NHQ6XYoNUT7isFRiUpKkvM+NEILy8nKUl5ejubk5caP1s58h9vjjcDY344duN/7+7/8elFIsLy+jo6Mj6/4+9rGP4eGHH8av/dqvZXrIuwEcSf53HsAXAJwnhGgBfB7AOwAsAbhCCHmKUjqW1wkJDgXyekf2nxwmHoFESRBL325vbyMWi3FzcKPRmDMrk42DFpOZGtvYiFV2ztFolNdsHz16lAc1Njc3sbOzcyhT2GrJduyEEO4o0tPTg+3tbSwuLsLtdmN0dBR2ux1Wq1XV+e+HQ4dcWP785z+H0WjkEUtWY1leXp4iLOMvvQTtiy8C990H7d13v36Mt2g6OxNvGjFJKcX29jaARNqPeREeO3YMVqu1qH17vV5sbm6iubkZPT09OR+faYGLxWIYHR2FVqvNKUjV7vOgkEcWNQDKr15F5TvfiYaGBgCJOj6Xy8VTrRUVFSmd1XLrnNjddyN+7hxw/XrRx1WqBUej0cBkMvH3W95R7fP5djW9ZHvOUorJYvejvXwZtb/yK0AkgjqdDv4nn0Tlb/82fud3fgcPP/wwlpaW8Ed/9Ef45V/+ZcXt7733Xh65zcCDAL5GEx/Wi4QQCyGkEUA7gGlK6SwAEEIeTz5WiMlbAJayZjfr8Xgcs7OzPGojryeLRCJ8TJ/H4wEALiiam5tT6q1XVlYOZQNMvrDuciYe4/E4qqqqYLFY4HA4cjb83ark89qzoRRGoxHxeBx2ux2rq6u4ceMGamtrYbfbYbFYMq6l+233RgjZFbFk9fZMWFYMD0P/wAMJ0fc//gciP/gBYv390Ol0MFy4AHILprMz8aYQk2wk4urqKoxGI7xeL0KhkOJIxHyQRxFZClcNSpFJVmfZ1taGpqamvI+FpYdWV1dBCFF9N7dXyCOLkk6HyB13QF5AYDab0dTUxFOtrPlF3sxj7ehAzZkz0Ov1oMna1mIp1YKjZJDLOqqB3U0vbIylkg1RKcVkMel2ALsmKBlfeQV3/t//N4xGI77//e8jFoshEAgU8xRNABZlPy8lf6f0+8M3hkmQYg4ejUYz1juyOdehUIgLKZ/PB51OB4vFgtraWnR0dGQdjlCsGMyn2bGUsFGMrFnmtdde4+IxXTDn4jCI4WIotM6SZYIkSYLT6cTi4iLGxsZgs9n4OEf5vg/SO1gesYxGo1hfX8f4+DjsX/86OsJhEElKuMG8+CIW7XYYDAY0NjZCc+YMtP39iVS40o7VprMLGbNYYt7QYjLdOzIWi2FpaQnt7e04duxYUaHzaDTKaxoHBgYwOzur+kufvkAy+6BsdZa5YPWabNze0tIS/H4/ZmZmEhNtMowX3CvkkcWZlhZUnz2L3dWoCQghvIZG3szjcrl480tVVRWPeBQjmPZKTKaT3vTC6hJnZmYQDAZTIrGlEpPxeLzo/WSboAQkjIqrqqqKeQqlF41m+b3ggFGqd2TiMb3ekd0YyscSsnrHQvwOS2HNU0gDYL6wKVXsnAHwGvKtrS2cyzKUIBuHKb1daANOsc+j0Wj40IZ4PI6trS1e311XV4fGxkZUVFQc+CAKcvEiNC+8AHLvvWi+cAHNzc2IB4OgX/86X0+XuroQDod5QCFnjaWadHaxYxZLxBtWTPJ6haQ32fLyMp/B2dbWpmof7M1O/4Cy5piOjg5uv5LPosXuluVTcdLtg9LPRX4M7EPLust9Ph/m5ua4nxqrnbt06RKqqqr4pBSDwQCr1VqQ6XghMFNu//g48nE6kzfzAIm7fKfTic3NTVy7do2PE1TbzCOnVAtOPgJQqS5xZ2cHLpcL4+Pj8Pv9mJiYULRVyodihTbw+k1A7Mc/xnJ3N1rOn0ckHC7lXOIlAC2yn5sBrAAwZPi9YJ9RU+/I1ip5+tbj8SAcDnPxWFVVhebmZv49LoRck8VysVdp7kxzra1W665o6+zsbMmffz+glMLn88HtdkOr1ZZ8/nY2Mq3RWq2WG4vHYjFsbm7ybFYsFkMwGFRsot1ryMWL0L/73UAkAq3BwIebaO++G7FnnoHmhRcQufNO0I4ObMzMYGNjA36/X7HGMkVYqkmFFztmsUS8IcWk3DsyHo9jbGwMGo0GXV1dKca3uUift00pxcLCAlZXV3Hq1KmUL1c+ix4hiQk7MzMzPDSe6cuTLmjTP7QrX/sapmw2tLa27hI38js6INFw5HK5cPPmTezs7PDomNVqPZAvoFpYc0t5eTluv/123kXN6hNNJlOKb2I2sVjKyGQxTQEsEtvW1obLly/DbrdzWyU2cpPZEKmdkV6ydPn58/D09CC0tQUA3D6oRDwF4OFkTeR5AB5K6SohZBPAEUJIB4BlAA8B+PelelJBbh5//HEMDAzAarVicnISnZ2d0Ov1KfWOzBycCUiWNaiurobdbk9ZRwKBQNFC7rCYhhcz17qY9WY/az7ZTS5zrQgEAlhcXITFYoFGo8H09DTC4TC3ZMpmPC7f515FM5kJemNjI6LRKC5duoSpqSk+zrGxsXHfrmuaF17IONyEucFokajlYY4ZkiTxueb19fVoaGjgEdYUYSlPhR/iZpw3lJhM9470+XwYHR1Fe3s7HA4H1tbWuB+UGuQCkRmaG41GDAwMKAo3tWKSmX2fOXMG1WNj0Hzzmxk9LNMXk/QPrfSTn6D/L/4CGxsbfFSe/PWQYzKZ+CxVeZ0i+0AzAVNTU6NawBwEBoOB350q+SZmE8mljEyWKrJLCEmxVWLRHrfbjZs3bwIAf29YGYMSpYhMKu2L3Xio4cMf/jCef/55bG1tgRCyBOCPAegBgFL6RQA/QMIWaBoJa6D/mPxbjBDyMIAfImEN9BVK6WhJTkagiu9///toamqCzWbja4k8fev1eqHRaLg5eGtra1ZBUYp6xYOKTMpHEzLnhpqaGjQ0NODIkSNvCBN/JSN0tnZ2d3djZGQEfX19AAC9Xo+qqircvHmTj9llwxHq6upKtu6w48p3bdXr9TAajTh16hQAYG1tDUNDQ6CUorGxEQ0NDXva5KR2uAm5eBG13/42dG97G6ruvx9NTU2IRqPY2NjAxMQEwuHw6807acIy/tJLMLz73UA0ChgMiYjlIWrGObyKIU/k3pEA+NSY22+/nUdV8l2Y2OPZKMOuri7eYKH02FyLFrsT8fl86OzsRPXYmGJoPNt+2YdWCocBvR71H/oQoNNlTMlnIr1OMV3AsM46Vh9VzOJZCgGXaR9KvonpIpmZjdfU1BR9LGwqj+7kSWhy2OQUCvOWYw1dzDpla2sLMzMzPM1fU1ODqqoq/t6U0q9SLiaZQFfDN7/5TfmPuxz7k13cn1ballL6AyTEpuAAMBgMmJqa4jW+165dg8lkQnV1Nerr67OagytRinrF/YhMshvS9LnWrM7T5/Ph9OnTBR/DYUFeu80ij6wxsKura1djYPo6SSmFXq9HZ2cnOjs74fP5sLq6ipmZGVRVVcHhcOxq/Cx0vS1kG/Zcer0era2taG1tRSgUwtraGq5fvw6NRsOFJbsJKlXUV81wE5ZVtIfDoH/7t4g98wzohQvQ6/W8GTWbsNS8+GIi6pls5ok99xyQtM3SHJC3pJw3hJiUm+BGIhGMjIygoqJiVwQxXzFJCMHCwgJcLhdOnz6dNdWXa+EMBAIYGhriNRIajSZraDzTfje6urD22GM4urYG4zvfCSQfzxZN+Zcjny9kuoBh6azl5WU+/or9/SCmLwDqzkepmcfj8SD8/PPYfvFFbB4/juDtt3MrknwujvKpPA16PXxf/CLQ3V3MKalCp9OllCuwNP/a2homJye5DRFQuoJ9uZhkFx3BwUEIaQHwNQB2ABKAv6WU/jUhxArgn5GwVpp3uVz8s6CWL3zhC3j22Wfx4osv4saNG/jt3/5tOBwO9PT0FPW+FxtVZPsotZhkooqlc/1+P59r3dLSssvK60CbOooQ06zMa2lpiUdYWT1rKabosHX2yJEj2N7exsrKCrfxcTgcBTfqFXO+6edjMpnQ3t6O9vZ2BAIBPs6RdVPbbLaSvb+5hpvw670kAdGo4vWeCcvmxUXQn/0MzhMnMJl0n2nr7ERb0tkEBgPi99wDKemZzZrgWPPOQXxmb2kxmZ7WdrlcmJiYQE9PD7/wytFqtaprJiORCDweD3Q6nWJaOx1mhaEEa4A5fvw4LBYLFhcXIUmSqtC4XCROT0/D4/Hg5Mc+lui4VXicHLZdIR8svV6P+vp63Lx5E/39/dwXknXRVVZWctPx/fBIK3SB0Wg0qJ2cRNknPwlEImjX6zH+uc/BWVWFmZmZvMzG5bY5AFB57Rrwi79Y0HEVgzzNDyRqYVkNaTAYhN/vTxmPWMj7H4/HeakDszYSHCgxAL9HKb1GCKkE8Coh5FkAHwPwY0rpo4SQRx599NG3P/bYY3ntuKWlBY899hi+8IUv4O6778Y73vEOjI2NHXiKGig+usmOQT5dht0cWSwWtLe3H9q51vkiL/lhIpmVde3lebIsFnOm2Nra4iVHZrM577Wj0GtWru3Kysp4VHVnZwdra2u4cuUKj16WMl2f3iQLpKXC9fqsqXCWsWw0GGB7+mlEBwawsbGBi3/2Z7BPTIDcdx+qT5xARfJmK10LybvC9+uzfcuKSTa7dHp6Gr29vZiamoLX6804EhFQv7i53W6MjY2hvLwc7e3tqlKHSvuWJAlTU1PY2dlJ8bRkC6Sq0DghCIfDGB4eRnV1Nc6ePZsx3Zu++JfyQ5TuC+nz+eByuTA2NoZYLMY7GS0WS0nrZxjFpKfTRWDtyAjKPvQhALvNxlkzj1IENsU2R69HoL8fhRk5pZ5XsbDaJRaJqK2tTRH+6YbwaojH4/wmIZ+aScHeQCldBbCa/LePEHIDiXr+BwHcl3zYV//t3/7tf+QrJt/znvcASHzHQ6EQgMORogYKi0yyawOLPHq9XiwsLBzoXOu9QJ6eT4+wMvF49epVNDfvqjbZMzQaDerr61FfX49YLIbx8XGsr6/D5XLBbrejsbFRVfBhL8SknIqKCnR3d6OtrQ1XrlyB1+vFzMwMKioqeMQy23U/22cyU2c3u95vPPEEKt7zHpgzRDGVMpa6CxfgcDgwf/Ys7J/6FDY2NjA5OYlQKLSrxhLAgQjLW05Myr0jmai5evUqWpaWcGxmBjQeVxRlQCIymW2BpJRibm4Om5ubOHPmDObm5vK2+2GEQiEMDQ3BZrPhzJkzu3yzWBQzV2g8FothcHAQvb29itFWhlKdSr51lGohhKCqqgpVVVVob29PmVs9NzcHjUbDo5ZFehKWhHQRGBwYACtYkJuNy+/s5UKMnYtJ5p250deH0PHjRR9bqescdTodryGV2xDJDeHZ2Dar1ZrR8ic9zS3E5OGBENIO4DSASwAakkITlNLVfFPcckwmE49kabXaWyYyySKPrJEkEonw+fLd3d0YGxvD8SK/qwflYZguyNn6xGo72c1vvjPL9wOW9amoqIDD4cDq6iquX78OrVbLaxeVGj1LmeZWe5w9PT04cuQIvF4vVldXMTU1xR0KrFbrrjU623Pl6uxeMZvR1dWlGL0Ecjfz6HQ63kgbi8VShGVdXR3sdjsqKyt3CUuXy4X6+vo9E5a3lJiUe0eytLbP58MdAKyf+ARvjVdqZAESi1umNHckEsHw8DAqKir4LFhCiOq0uHzR29rawsTERMZRjWqLwtn0lNOnT6uarqO06O6HrUT63OpIJAKXy4WVlRVMTEwgFovBYDBAq9XuKvJWSzGLudxAffXoUYQyFNNnauZhfpCRSATVFgtqPvIRRCIRkBK8tqUUk0r7UqohZRfe0dFRxGIx3qBksVh4YXqhDTiCvYUQUgHgXwH8DqXUW8oLgslkSolM5mOjpkS29VYtSmul0lxrNl2mt7c3JTO1X7Y6e0UkEuHd5KwxqKamRrG2c68p1OaHEAKDwYC2tja0tbUhEAhgdXUVly9fRllZGe8Il1vwFXpehR4f25bV0lOaGL+8urqK8fFxWK1W2O121NTUpARpcqazFcSgJEnQX70K/Qc+oNh8qyZjyVASltPT0wgGg7uE5eTkJKxWa8r1uJTcMmJS7h3JuqKj0SjKyspQ/fOf52xkATLfKbtcLty4cWNXrWU+d9Zs4ZyamsL29jbOnTuXMZyfa79sRrdOp8saPZKTKfV9EBgMhpRo3/DwMABgenoaoVAopd5SrRl2sZEBZqAeXFqCpoCO97a2Nt7M43a7sb6+zhu+cln2ZD2uPerAzgSzdqmuruZRZXZOCwsLoJTCYrGkjEzc2dlBfX19zud/5pln8JnPfAbxeByf+MQn8Mgjj6T8nRDyXwD8avJHHYBjAOoopS5CyDwAH4A4gBiltLCRIW9gCCF6JITkP1FKv5389TohpDHp1dnIZsUXgtFo5GLyMNQ7suNgUZVC5lqXYg3cqwyPEqFQiGd5fD4fD2o0NzcnOnrfAJZEZWVl6OrqSukIn56eRnV1Nbet269rF83gF5xeB8qCI2ycY11dHapGR6H/z/85Yzo7kxiUJAmGl1/OqlmUMpa5boxyCUsWhEtv1C0Vh15MpheW+v1+DA8Po6WlBc3NzXjllVdUezylN+BQSjE7Owun06lYa5krLS4nHo9jbW0NTU1NOHfuXNYvQ7bI5M7ODoaGhviMbuaVlYtMNZP7dWfOLHNid98N6fzrI5UJIdDpdNyQlUXGXC4XlpeXIUkSr7csVJDlQzELlUajSZlSQymF2Wzmlj0srcMm86h5nr2OTOZCyYZoe3sbm5ubmJiYwJe//GV4PB4MDAwgEolkTYt/+tOfxrPPPovm5mb09/fjs5/9bB+ldIw9hlL6FwD+AgAIIe8F8LuUUpdsN2+llG7ld9ZvDkjiw/RlADcopX8p+9NTAD4K4FEAH33wwQcLfg6z2QyXK/F2HKRHJPsMbm9vY319HRqNBjabraC51qVgL9dRVrPNfDwNBkPK6Emv14uNjY1DUS5UCNnWW3m5VE9PDxdsTqcTer0eXq9X9Tq6F8fH0Gg0qJueRsMLLyB+zz3YsFqxsLCA6kuXAPncbQWjcmB3Mw6lFPF77gFUaBY5+azvSsIyHA7jlVdegc1mw7Fjx1TtJx8OtZhM945cXFzE0tISTpw4gcrKytcfpzIsLF/cWFNLVVUVzp07p/gmqV0MWRc5s0nIRab9rq6uYm5uLuX81C5k7HHb29sIh8NcHOSzCGYShGq2Y5Y5BoMBgaeeyri9PDLW0dHBLxxyQcbEjTyNU6q71VLtR5IkGAwG2Gw22Gw2AK9fGJaXl+H1emE2m3kENlPR/35HJnOh0+lgs9mwvLyM3t5e/Mmf/An+6I/+CJcvX8aFCxdw+vRpfPnLX9613eXLl9Hd3Y3Ozk4AwEMPPYShoaEHAYztenCCDwP4Zoa/CXZzF4CPABgmhFxP/u4PkRCRTxBCfh3AQno0OB/kNZOlSFGrXT/T51qzdKPFYuHZATWR8b2iWDEp35bZerHGIL1ej5qaGjQ2NqKnp+cNEXksBEIIL5Xa2NjA0tISr1tn02xKOIWLo2b4RHpDTcPTT6P69GnM33kn6BNPAJEIJJ0OC52dsCTtlzJtG336aUgaTV6pbEamKGoudDod7HY7FhYWcO7cObhcrj0R6IdWTMqbbFja12Aw4Pz584oXzFyNLMDri5vT6cT4+DiOHj3KhUC2x2eCNexsbW2ht7cXGxsbqs4tfXGSJAkTExMIhULo7+9PmSqRz0LG7m7rZ2bg/PnPoevtxaLFgrq6upz1NfkIwnTk3dI0EoHupZcQUbttUrzIBZnL5eL1ouXl5bBarSWzHyqlKE3/Yis186TbKbHIJTufg45MZoIJ08bGRlgsFnzqU5/CfffdB6/Xq/j45eVltLS8Plo72UHapPRYQkgZgHcBeFj2awrgR4QQCuBLlNK/LcmJvEGglL4EINMH923yhxb6HCaTCeFwGEBpIpOZ1q70udZarRbV1dWKc60DgUDRqfJiKUZMRiIRxGIxTExMwOv1QqfTwWKxwG63qxKP+5ldykUhx1HIeqvRaFBRUYGenh7EYjGsr69jdHQUkiTxjvBSRadVRSYVGmqkU6fgP3mSz92O3nUXSEcHr61nArhCYVv6lrckvl8qNIucYtb3eDwOjUbDr7d7waETk+lpbY/Hg7GxsazTZ/IhFAphdnY2q4UQI5uYlDfsnDt3Dn6/P6/6SvbYUCiEwcFB1NfXo7e3d9cHW81iEo1G+UzSgXgc5k9/OuGpqNNhweHAQtJXi3UmK4mzYgShvFsaBgNid9+tajsljEYjn7dKKeWjJ1dXV+H3+zExMcEtiNTMhk2nlJHJXKUM6V3VrPGF2SlVV1eXTCRrLl1Cw7/+K0zveQ+gImWSC7b4AKkNOJnSbRk+o5k+uO8F8PO0FPddlNIVQkg9gGcJIeOU0hcKPHxBAZS6ZpLtQ57K9fl8XFDV1dWpmmt9GMSU2mNQEsqSJBU0Qegwsh+1jPLXWqfTcUu6cDiM1dVVXLt2DXq9Ho2Njaivry9q9G/69UBtQw3bjglCDQAHAIfDgWg0ivX1dYyMjKC8than2HUquW2horAYMVnKQEMmDpWYZF16TOTNzs5ia2sr5/QZNQIhFArxRpBcNY2MTKkeNl7xyJEjPP2Sz+LLFkgWIc3U9S1/bCZ8Ph+Gh4dht9sRCASg+973uCgklKJ+bAy29743pTNZyRuyGEEo75bON0WeDUIIKioquM3H8vIy6uvrebMIgJR6SzVfloNKl8vrg9ra2hCPxxH66U8R+dGP4DtyBFdlRuOFTuZpDYeBf/gHBL773ZK8B0piMhPNzc1YXFzkPy8tLQHASoaHP4S0FDeldCX5/w1CyHcADAAQYnIfSY9MxmKxgvbDvA83Njbg9Xrh9/vzisbJKYWoLZZsx8tS9G63m4vHdKF85cqVvKcSHQYopfB6vXC73dDpdKo9atMppsNajtFo5NNs/H4/7wgvLy/nwYd8SenmzuEPKReZ0s5Oxs+FXq9Hc3MzmpubET55Ess2G2LPPQfP6dModzgQX1go6BokxKQKmHdkJBLBa6+9hrNnz3KTbmbTkwm22GS7+DKrnt7eXkxMTKh+I9On2lBKcfPmTaytre0SuPmkhVjE1e/3Z+36Zo/NtF9WY3ny5EnE4/FESujee3lhL9XpEL3rLmixuzOZeUOyNKzOYEDT3/4t6kZHoX3b21LEiBrhxLqllSiVgJM3wACvL+RswhCzzbBarRmnPZQyMlnMl1N/9Sqqf/VXgXAYzXo9dp58EltVVQU18/CosiSBRqN5RZXV4Pf7U2qUlejv78fU1BTm5ubQ1NSExx9/HEg0h6RACKkG8BYA/0H2u3IAGpow4y4H8E4Af1qyExCoIl1MqhFxrJxDbpzN5lrX1dXBYDAUVexfio7wUsDW4Gg0yt0PWH2nxWKBzWbLGWUthP2MzFKaGDPpcrn47O6lpSVuhzM9Pc076RsaGlSda6HHnmuNLi8vR3d3N7q6uuD1erGysgK/34/R0VE4HA5eb6vm+NjjcvlDpnRbq7yOGI1G1D/4IPDgg7CGQlhdXUUwGMTVq1d5ZFVthk1NfWe2beXXqzdkzaQ8rc3uhl999VX09vZy38JssA5tpQ+2JEl8BGEu0aaEfEGNRqMYGRmB0WhUHK+odtGLRqMYHx9HPB7HhQsXVI1pVJqsMzk5iWAwiIGBAeh0Ovh8voRt0vnz8D/5JPQ//zkmGhvReO4clL7y6d6Q4XAYLrsdY93diZT46CisVuuhWMgB5UVJr9enzKxmhr5slJdSWv+wNPLIBSBiMRhfeQV1d97Jz4WlBpeWluDz+VBWVpZxRCKLKkvhMEiRZQZKqIlM6nQ6fO5zn8P999+PeDyOj3/84xgcHBwlhHwSACilX0w+9AMAfkQp9cs2bwDwneQ56QB8g1L6TElPQpATNWKSlZ6wVK7f70dZWRksFssu42w2haYYDjrNHYvFEI1GMTc3h52dHS4erVYrOjs7VYvH/bS7yQf5+EVWo15TU4Pu7m6MjIygr68PQGKtLS8v56Lt4sWLqKqqgsPhgNVqLem55fN+y70hXS4XGhoasLS0hLGxMdTV1aGxsTHrjbC89l2tKwy5eBGmZ59FZVcXcPKk6mM1mUzo6OjA6uoq+vr6sLq6iqtXr/LSrrq6uqwp+1JFJvfqc3igYlLuHclMNaPRKC5cuKBa+GVa9NgEmtraWtVp7Uz79nq9GBkZQWdnZ8a6TaXjSK+/YPtxOBzweDyqPhjpi2k4HMbg4CBsNhuOHj2acl7scfGBAdALF7AzPKz6i5mpVjEUCvE0zV6OS8yFmsXYbDbDbDZzr7L0tH51dTVCoVBJbDaKjUymT+VJF4DpzTyBQIBP5gkGgymTeYzJMoO1xx+H/aGHSlZmwAgEAqo6KR944AE88MADKb+TiUj28z8A+Ie0380COFXscQqKQ6lmkn2PCplrXcq6y/0iFovxyCMTwswUvbu7u6j6vEIppZhm3eRs4AfL5ihN0FF6Xw0GA44cOYLu7m5sb29jZWUF4+PjsNls3M4o/dhLlebOBSGEN3PG43Fsbm5yb2PWEJOeqpdH+1SNN06mwvWRCHp0OsQ6O1V1YqfDvDa7urqws7PDs4zMxN1ms+26zhbbgMP2t1c3ZwciJllaOxqNghCCYDCI4eFhNDQ0wGw25xVBTPeOBIDNzU1MTk5mrEVU+2ElhHDbmrORCMq+9rWMH7L0RS+9/mL5q1/FdF0dTp06BUII3G63qvOTi0lWq6nUhV7KO3h5reL6+jpuv/127g05OzvLU7C1tbWHboQXQymt7/F4MDMzg5mZGSwuLqakkfP9khYrJlmdaeDpp7Fz9ixsWQQgIQTl5eUoLy9PaeZJEco1Ndj64AdhPXu26C91+ueIUnogF1HB/sJmcw8NDcFgMMDlcmFra4vXLHd1deU1waoUa9JeRyZZuQ/7j1LKJ0K1t7dDp9NhZGQEFoul4O8AO4dCBVIxUEqxubnJ0/Js7WbCL581TP4+kDRTb1ZqFA6HeVCimObCYs9bq9Xym3F5QwyllI9yZF7BKQGZHB3W8lQ4+zmfjmwlKioquECXm7hXVlaisbERtbW1XF+ImkkZcu9IQghWV1cxPz+P48ePo7q6Gqurq3l98eRiUpIkTE1NYWdnB/39/Yr2AWpqLIHEHer8/DzC4TDu0mhgzDD6KH2//Oe0+gs8/zwG/uIvoNVqEQwG86qvlCQJCwsLWFlZwZkzZxQLoZUW3VIt5unG1iwFu7i4CJ/Px+17rFZrzg75Qik2TcTOYXNzE3a7HWazGW63G6urq5iYmIDJZMrpCVnK4wESgtKZsNDJC3kzj3yKzcbGBgYHBwG83phUVVWVdyQ5/S72MHTTCgBCyFcAvAfABqX0tuTv/iuA3wCwCQDf//73d0WHczEzM4MnnngCP/zhDzExMYHPf/7z+P3f/31UV1ejr6+vKJP/wxaZlE98kotHlqZXql8rxTq6X98h+fmxukev17tnNZ1A4j1iwi0SifAZ3DqdDjqdTlW5mpxSv1byhphQsm7x1VdfhdFoREVFRcbny9XZTXU6VWbjapGv6z09PfB4PPz6ZLFYVF2XMvGGE5PytHY8Hsf4+HjCziZZ9we8Lp7UfuiZ9UIwGMTQ0BDq6upw5syZjC96thpLBptCU1dXh0gkAt33v59zXOMu78jkh04KhwG9HnUf/CCQfM58FkjW9FNeXo7+/v6Mx71XYlKJ9BQsS4mzEZfMM66mpgZarbYkx1CQX5mCCTvbj8FgQENDAxoaGngTgTyNLPeEVLopKdWXU5KkoqN+TCgbjUacPXuWNyaxNA8zgq+pqVE1UULp+3EYo89vQv4BwOcAfC3t939FKf2fyX/n/WXzer1obW3FX/3VX+H3fu/38KUvfQk+nw9LS0tFve+lGslYzPoRj8cRi8UwMzMDj8eDeDzOI4+ZxGOpj6GY1zDXc8snirndbn5+VqsVbW1teO2119DV1VXw8+eLfAY3myk+PT0Nt9utur5yL+tLWd1iR0cHdnZ2MDMzA6fTiVgslhIFzNXZHXzmGWz29aGlyKhkJlhtrsViAaWUX5u8Xi8opbDb7aqbjIA3UANOunfkzs4ORkZG0NraiqamppQT0+l0eU3y0Gg02Nrawvr6Ovr6+nJaMORa4JaXl3Hz5k2cOHECADA3N6eqMDf9zVnv7MT6Y4+hd20Nhne+E5B96NQuToFAAMvLy7DZbLjtttuyPvagCtXlKfHW1lZ+Z+xyueD83vdgHR6G7rbb4K+vz/vOSi4GoWKyUPq2SibsSguV3BOyqakpJY08MjKCeDyeYqPEBPJh6ApXIr0xiRnBpzfzWK1WxbTlftTXCPKHUvoCIaS91Ps9ffo0Tp8+DZ/PV/LZ3Pud5pYkCR6Ph3eXMzFZUVGBlpaWgsyuSyEmS/U9YjWsLPIYDodRVVWFmpoaNDU17emoyXzPoby8HDabDU1NTTCZTLy+sq6uDg6HI2tT337cvFZUVMDhcMBkMqGhoQErKyvcx7j7Rz+CPktnt6+rC0GVpWrFQgiB1WpFJBLBzs4OqqureZORzWaD3W5HVVVV1tfsDRGZlKe1AfB07cmTJxU/TEo1kJmQJInXHmZKa6eTaZGMx+O4ceMG4vE4j5Tu7OwkitDzGH0kT7Wf+NjHoDcYdoUK1CzUrO6zoaEB1dXVOc9rLyOT+eyDRclsU1Mo+4M/ACIRtOl0GDUaMdfVpTolni4GDV//OkhyVJ8aMpmwqxGBSmnkFBslnQ6hUAh+vx8mk6moha9UX/Js71F6cxVr5mHF6SnNPEZjipiMRCJ7VrogKBkPE0J+DcBVl8tVsKeh0WjM2xooG/uxD9YgydLWsVhsl7h69dVXFRsa8uGgbqpYIGZ5eZlbL1VUVKCmpgY9PT0Fez/uF2y9ZWs+a4yZmJhANBrl65L82r2frzWlFOVDQ6idm0PNvfcifuedcDqdWOzsxFGdDhoA0Ot3BZDkXeD5Pl+hsIwtCxTE43FsbW1hbm4OgUCAd68r6ap8sr2FsqdiMhQK8WHt0WgUo6OjMJlMGBgYyHhiasVkIBDgheIOh0P1HZnS/v1+P4aGhtDU1ISWlpbXfadk3pFqRh9JkoRXX30VVqs1a6o9m8ijlGJ2dhYulwv9/f1YXV3Na0FO78YrRWSgEORiTkMpOhYW0P7hD2dMiacXuKeLQdPFi0AeYjKTCXshEUUlG6Xr169jbW0Ns7Ozu2Zw50MpxaTapjJ5M48kSSld72WDg6gbHYXx9tsRO3IEOzs7qs/pmWeewWc+8xnE43F84hOfQPqcaELIfQCeBDCX/NW3KaV/mvzbuwD8NQAtgL+jlD6q8tTf7HwBwH9DIr39337v934PX/nKVwraEcsKAYc3MsnSuiwyxzqtWVOJUtPHQaapC3l+lklg87vj8TgfNVlM3VwpKLaJSN4Yw+or2USbpqYm1NXV7auNkv7qVTT9xm+ARKPQGgzA00+j7sIF1P2H/4BoZyeCTz+Nxe5uuCiFfW4OjY2NMJlMBXs+FnNu6dcKrVbLS7VisRg2NzcxNTWFUCiE+vr6lHnm8olme8WeiEl2N+X3+zE7O4u2tjbcuHED3d3daGhoyLqtGjG5traGmZkZHD9+HNvb23kteumLJBMErAEo32NhsGLn3t7enLMvMy3U0WgUw8PDKC8vx9mzZxO1GyoXIvY4r9eL7e3tvIueS41czEk6HaJ33gl9lpT4/Pw8NBoNv4O1pInB4PnzeX0JM03lKcVCZTQaodfr0dvbC61Wy41+WaSvqqqKi8tcNVmlEpP5lIbI0Wg0PArbub6eiCYnjdS/du0a/ufPfw6tVouf/exnWS274vE4Pv3pT+PZZ59Fc3Mz+vv78dnPfraPUjqW9tAXKaXvkf+CEKIF8HkA7wCwBOAKIeQphW0FaVBK19m/CSH/5/Lly39Y6L7k34tSicliIYQgFAphfn4e29vbiEQivJ65r69PVcfwYUhTZ9s+Go1ycezxeGAwGFBTU4Pm5mZoNBrcvHmTzbq/5ch23vL6yp2dHaysrPCBDVVVVXmt1aqukQoNNcZXXgFhAx/S0tnkzjtRduedOIpEhmZtbQ1DQ0MghMBsNheUsSk0osm2zfR66HQ6HuWNRqPY2NjgTh8NDQ2Ix+MwmUx7GvXdEzFJCOEXN5/Ph+npafTHYjB/9as508TZBJwkSRgfH0c4HMbAwAD0ej18Pp9qwcf2L0kSJEnCxMQEQqEQ+vv7FS/6ahZU1iCzvr6OsrIyVUPUlT4QbCxiupdlPmIyGAzixo0bsFqtGBsb42n6xsbGvEf0FYtczE06HKg/dw7pr3B6l3gkEuG1fTcANPx//x/s4+PQvf3tCPX2guzs5H0M6ZNgSnXXy/Yjj/S1tLSkpN2WlpYgSVLWMYmlbOQpdj9yI3VNLIZ/73DA8Zd/iT//8z/HE088gd/93d/Ft771LXQqRIgvX76M7u5u/reHHnoIQ0NDDwJQIwgHAEwn/SZBCHkcgNpt39QQQhoppavJHz+Qq7ZaLQc1xpBFylnamjlfNDQ0oLe3t6AL+EGLyfT1hnlZulwu7jdssVhQX1+PI0eOpHyP/X7/m6JuuaKiAj09PThy5AgmJiawvb2Nl19+GfX19XA4HCgvL8+6fa51PVNDTfD8eVQbDEA0mtWo3GAwoLW1Fa2trQgGg5iYmOA16MxwXM31tdgpNmqeg0V5m5qaEIlEsL6+jsXFRa7L6uvr96Q8Ys/S3JFIBMPDw4kaxHgchve/P6u1DiOTmPT7/XwG9bFjx/gbotVqeZ2PGjQaDQKBACYmJmC329Hb25vxzc21oMZiMYyMjMBgMKC/vx8XL15UfRxy5GMR0+sdWHd7NpgwZobv8XgcHR0duHHjBsrLy/mIPoPBwMVbPumSQhczJuZ8w8NoUPFcBoMhtUv8+HFs3HUXXC4XAlNTMBgMqKysLMrzba8bZ9iFwWKxoKOjA7FYjHuVzszMQK/X8+aXioqKA49MylEyUtdrNOjp6cHnP//5rNsuLy+jpaWF/5yMpDQpPPQOQsggEnO7/zOldDT5uEXZY5YAlNZ5/Q0AIeSbAO4DYCOELAH4YwD3EUJuRyLNPf9Xf/VXRT8P+47sh5hUaiiR1wRKkoT5+fmMwyLUcNBiEkh4BLPzpJTyhr6Ojo6sa9mt7qKQ73rLmiHLy8vhcDiwsbHBS6KU6ivVPk+mUYmh06ex+Pd/j6apqZyBLobZbIbNZuPrOIuoqpkIVExkUpIk1aMXGQaDAS0tLYgkew/i8Ti8Xi8sFktBx5CNPROT09PT6OrqwsTEBLQvvpjTWoehJCZXV1cxOzuL2267bVcqOt876GAwiLW1NZw6dSrnC5pt3yyS2N7eDofDofr55cijo3J7pPRjiMViGfcRCoUwODiIuro6+P3+lNdPo9Gguroara2tABLnzppIgsGgqnTsQS1m6V3ia2trPFqRnhJXY3fDKGU9jpr96HQ6PpUBSLxfbrcbCwsL2NnZQSwWg8lkgl6vL6rRpRSilEWT/d//PoLnz8N6/jx2fvrTnFEBIOMNR/ovrwFoo5TuEEIeAPBvAI4AUHoh3/jhmDyhlH5Y4ddfTn9YoftPT3PvBUw8sm5rZsNlsVgUG0qCwWBJmnj2U0zK3SBY3aPBYIDNZkNLS0veguCwsJ+1jCTpb8wEZDgcTvGHdDgcqK+v559T+bHl8oeURyAppQifOYP4Bz6Q1/Gx9bayshJHjx5FT08Pnwh048YN1NbWwuFw7OqyPijjcUmSYDQaucPHXrBnYvLEiRN8EVA78xJIFZPMizIajfK0drbHZ4N1Wft8PnR2dqpS5pm+OCsrK5ifn8eJEycKHh8lSRKuXr2KlqUlNE1Pg4bDyuOblLq0k18Wz+nTGCwrQ29vLywWCzY2NrJuazab0bK0hI6XXkL0rrvgttvhcrmwuJgICjFxlstm4CDQaDQwm81ob28HkJoSz8c4fT8XRCVMJlNKZ/X169d5+UYkEknx6Mwn+lqqbj3p/Hls2O38NdzZ2ck5lxtIRCLZ5wgAlpaWgET0kUMp9cr+/QNCyN8QQmxIRCJbZA9tTt9WcGsijzyycYws8tjd3Z1zok6pmniKEaS5joF57bLoqtwhoa+vDxMTE+js7Cz4ZvEgO8k3NzfhcrlgMBgKrhHMd71VOl+j0Yj29na0t7djZ2cHy8vLmJmZQXV1NZqamvhYz1z+kOkis9DrQXqEkZDUiUBbW1uYn5+H3+/nzTDl5eVFp7mLEZO3dDc3Ix9rHSYOd3Z2MDw8vKvDOtPjsyGf082KmgtBftFXiiSyRSfXh4U165z0+2H79V8HkiJbKf2fvhDyL0s4DItej/6nnoKhtpbP0VU6Hka63Q556ilYkjWF0WgULpeLe22VlZXBarUeSN2UEumv666UuKxLPBKJpPhCyt+ngxaTcggh0Gg0aG5uhtFo5B55LpcLCwsLAMBTKVVVVVk/t6Xs1pOnzNnFPxf9/f2YmprC3Nwcmpqa8PjjjwPAU/LHEELsANYppZQQMgBAA8AJYBvAEUJIB4BlAA8B+PclORlBXsjXsEJEDPsusshjIBDA/Pw8ampq8h7HCByOrnKl7dmQA7fbjZ2dHZSXl2cUyMU8/36uVWz9SZ+eU19fDwCYnZ1FOBzmQyv2ytMy1xpdUVHBo4EsoMAM6WMvvpjVHzI9I1qMxU+m7TQaDerr61FfX49YLIaNjQ3cuHEDsVgMVqu18LKxImdz35Ld3MDuL4Eaax0gIQ43NzextLSE2267DVVVVTkfn01Mbm1tYWJigs/pXlhYyKthh8Em7DQ0NKTUbMrJNT+TUoqFhQWsrq4mjKNffjnvyTrkZz8DwmHeJGG+dAnxt7wl44Il/10m70UgUbQrnwjDxFkgEMC1a9e4qGGm3ftNtgUmPSWerUs8Ho8fGjEJpC4QGo2G390Cr3d5rq+vY3JyEkajkUct2Z240n6KRS4m2YUyFzqdDp/73Odw//33Ix6P4+Mf/zgGBwdHCSGfBABK6RcB/DKATxFCYgCCAB6iiQ9ojBDyMIAfImEN9JVkLaVgnzEYDAiHw6oL9OW+pdvb2/D7/SgvL4fFYkFnZyeCwWDOgQvZ2At7oUJg3bxutxs+nw9GoxE1NTVoa2vb9V28VZCv86xelWVGWlpacP36dXR1dYFSCqPRCEIInE4nt8Azm81oamriU2MyUaydULbHMMu2QCCA1157DdPNzTjO/CGzZEFZds9w5AjiAwN5H5/aCKNOp4PD4YDD4UA4HMbNmzexvb2Nq1ev8hnhajNQYjY31EeD4vE4lpeXEQ6HceHCBVUvcqY7V0opZmZm4Ha7ce7cOW4hodVqEY1G8zp+JkhzTdjJdhcdj8cxOjoKjUaD/v5+XLp0SfVkHbYQhkIhzNpsOKWi+4xtKyeT96LSdkycOZ1OHDt2jC86s7Oz0Ov1XJwdxoVUqUs88OMfQ3r+eeg7OnAjKcrYBBi1pEzkKfKLyfZVUVcHzcmTio/R6/X8Dhd4veaVpU/kIx9L0YDDkKdE/H6/6prgBx54YNdc6KSIZP/+HBLjAHdBKf0BgB8UeMiCEsGMyzNFECmlKVE5v9/PJyq1t7fvWg/UZmsyUWy9o/wY8oGNJHW73djY2OA32w6HA5WVlXldmIsVs6VMczMPS5fLxW8UrVaragN0nU6H9vZ2tLW1wefzYXl5GRMTE6irq0NTU9OuLEah0e180Wg0MJlMOPqxjyHU1YXAD36Axa4uRIxGONbXUVdXx98zeSq8Ra/H2te/DjQp9QpmP8Z8P9NGoxENDQ2IRCLo7OzE6uoqLl++jLKyMt4Rnu1zVSoxuVfX6z0XkyxymEsYsrS2xWJBZWWlarWuFJkMh8MYGhqCxWLBuXPnCvZPo5QiHA5jbm4uRZBmItPCFwgEMDg4iJaWlhS/MDXpf7YQud1ujI2N4dgHPoDY8eO7tlH6gKQvYpm8F3ORbtodCoW4qAkEAqisrOTiTKmutRSp5WL2YXrtNVg/9jEgEoFDp4PriSewabFgcnIS4XA4Y0pcTnqJQNVjjwH9/QUdj3xfp3Q6hI4eVfVesEgAG/koNxsPBoMwGo2oqKgoqtsdSLgUsIVHbZpb8MbAZDLxkYrA6+KRCaudnR2YzeaM4jEdtt4WeqNTiq5yNWKOZTOYSCYkMRvZZrNBp9OhoqKC39QVwkGluSml2Nra4jZEzE2itbUVFRUVqiOAbF+8yYW8PiVMkiQ+sS0SicDhcBSVBi90rWfb6O65B1X33IPjAII/+QmC3/wmRrq7ob3rLjgcDtTKOrsBwHzpEvCe92TZ824KFXZsu7KyMnR1daGzsxM+nw8rKyuYmpqCxWKBw+FATU3NrtfgTRuZ5G9scqpCtoubfB52LBbD6upqxsemky4mXS4Xbty4gZ6eHsXOJbUNO8zaiFK6S5BmQmnhY18ypU50QF36n6Vsz549mzAezbGN/FjTFzEl78V8MZlMPHRPk0bpLpcLy8vLkCQppZGnVB/gYsSkPL1PKEXlq6/C9Na3cl/I9JS4vE6Rf47TSgRqBgcLPpf045GXG6iFEILKykpUVlaira0Ni4uL/KKffh75RlLSI5NCTL55MJlMGB8f5x6Ply9f5uIxHwHCOAy2PEr7YJN0WIo3Ho/DYrHw1LX8euXz+Q50gk4+MJ9bdl6BQAAejwc2mw1dXV17UqKk0Wh4iZS869psNoNSmvd4z/S1Xqk7O9c2bLvqX/olVEcisBsM2Hr8cczH41iyWHAmGfSgej1CFy4gv7llxTXupEfumSinlPK+hRs3bsBms6GxsZG7lRRrK3TLN+BotVrEYjHFqF4sFsPYWMKXmDW0+Hy+rFY4SvuPx+OglGJubg6bm5tcdCmhJjLp8XgwMjKCI0eO5O1hyfYtT7Nnmhue6wMZj8exsLCAUCiEO++8M+8PQ6kWsWwLKSEE1dXVqK6uRkdHB6/zW1tbw+TkJMxmM4LBIEKh0IHNeE7xT9TpEL/nHv639DrFSCQCt9uNlZUVjI+P80ak+uR7yEoE3KdOodA4RfrxZCo3yJfKyko0NjbuOg+fzweTyZSS2s/22YjFYiliUk3NpGB/IIR8BcB7AGxQSm9L/s4K4J8BtL/97W/HE088kdcFnFKKf/zHf8RPf/pT/PznP8f6+jo++9nPoqKiAufOnSvqIlRsA00p1jB2DPIxjOFweNcM72zHcFCCWE0nOZvAxTrJmeWbvO5xv2Bd1ywNPjIygrGxMdjtdsU0eC4ydWeno/QapXtL1gwOour++xHr68Oaw4HIs89i7ehRGNrbURmN5mXZVKiwyxYhlNeAsmjv7OwsAoEAT4+XogHnlk9zp8N8GltbW1NSv/mMMARe92G8du0a6qanccf8PCilWU3RMy1ulFIsLi5ieXkZp0+fRllZGWZmZlTfhbBFKxqNYmhoiC/G2VLQmfYbCoVw/fp1WCwWmEymghb0Ut3V54O8zo8tdCMjI5idnUUsFlOVUlaimMikPL0/XFuLo1migAaDIaURiS3UY9XVMD72GBrHx0He+lZ48rzbLvR41BKPx1Nu2NLPg9VbzszMcG8/1syTfiEVkclDzT8gUXf6NdnvHgHwY0rpo48++ih99NFH8dhjj6neISGJ6Vm/9Vu/hUgkgt/6rd/CyZMnce3ataLXj4OapCOv7dza2sLm5iaqq6u5GXo+tdKlWEdLSTgchtvt5nWPZrMZVqsVR44c2ZPJJoXAIm5Wq5WLI7VpcPlan8lsPB3t5ctoevJJEJnFXqaeBJ1Oh9r3vAd4z3uwneyyvnr1Ks+45apdBAqfZKNWhMqjvbFYDGtra/B6vRgcHERTU1PeZQRvmDS3PNJIKcXS0hIWFxcVJ75oNJq8xKTX64Xf70efx4OGT34yq81O+v7l4fNYfz9vkBkYGOAX01wd2un7ZrWfXV1dWeeQZ1ugWKq+r68PhBAsLy+reSkUOchFkJDEqEGz2Yyenh4YDAZsb2/zlLK8USbf9Fm+sPS+5/LlvI+fj0o8cQIejwdOpxPB5eWULvd8vTnZ8XivXCnkdHbvL8cdb1lZGcrKytDc3JyS4mOlCSzFZ7FYUvbFamJz8cwzz+Azn/kM4vE4PvGJT+CRRx5JP4ZfBfAHyR93AHyKUjqY/Ns8AB+AOIAYpfRcQS/CmwBK6QuEkPa0Xz+IxGQcfPSjH8V9992Xl5gEgN/8zd8EkKjLZdmYUgjB/RSTbCBAem1nTU1NyuCAfDkMkcmtrS3eMa/T6fjs7nwGNhRLoTf0Go0GdXV1imlw1g2uvXz59VR2bS1/HlVNqhcvovL970dlJAL8wz/k9JaUo9VqYbPZUFdXxxuKpqamYLVa4XA4UF1dnbERrZDXohARqtPp0NzcjLW1NRw9ehROpxPXrl2DXq9HY2Mj6uvrcwZm2Jq+l9Z4+xqZjMViGB0dhVarxfnz5xWjbazGMhfpNjs2FTY7wOuLW3r4fOQv/gLWd70rJUrKjl/tYsgsMk6fPp0zmqO0wLBzWltb46l6j8eT10LkdDqxtrZWlJ/VXsA8FeVd1qyzkE2DkTfypN91FVMvUipYSryqqgperxe33XabYko83y7xUpBPNzebjMRKE9jIR9atHwgE+Lx5Zgif67k//elP49lnn0VzczP6+/vxvve9D319ffKHzQF4C6XUTQh5N4C/RerIxLdSSrfyO2tBkgaanM/d2Ni4a3hBPphMJgSDQQCHw+MxG6ykxu12w+PxwGAwKIqs2dnZop5nv8Wk/GbP6XTy7IDVakVnZ+eBWLMVSrp4SU+DLy8vY/U738HZ//JfQGIxaA0GmP/2b4G3vCWxvQpBqHnhBSAaBZEkVd6S6cfHriuVlZXo7e0FpRROp5NflxoaGtDY2IiystcrK4ttwCkESZJgMpm4ebvf7+cd4Wz8ZCabpmK8Y9Wyb2LS6/ViZGQk5/hBNWlu+UzsgYEBXLx4UfWUHbZ/efhcCofRs7oKfZqQBNRFSuVjEY8dO6YqLZi+UMfjcYyNjYEQgv7+/pT6BjUfANZ5Pjs7i+7NTeCrX8VGZyc2T5xAMBhUFf2TW9+w7mK1z6+0rfzYlDAajXwajObSJcT+7d+wdfw4RpKNMSzqV11dfajMxtndZaaUOOsSL3SaTaHHVOgilT7y8dKlS9Bqtfirv/orDA8P4+GHH8a73vUuPPDAA7tutgDg8uXL6O7uRmdnJwDgoYcewpNPPpkiJimlL8s2uYjElBvBIYNZAwGHLzLJbnqYeNRoNLBYLGhoaMCRI0eyRuaLvYju5UVYXobicrkQCoX4jXV9fT2mpqbQ3d29Z89/EMgbTzRPPQUSi/FrsfbFFxGX1ZHLBWHGUYl6fWKOaA67vHSUIoWEEL4exmIxrK+vY3R0FJRSNDY28kEZe5nmznSs8m3Ly8vR3d2Nrq4ueL1ePnDEarWisbERFotF0aZrr9jzNLdWq8X6+npi4otCWjvTdplgorSjo4M3GwDqp+ywxS1+zz3Q6HQApSBGIwzveIfiUNtciyGbrlNXV4f6+nrVHzD5foPBIAYHB+FwOPgcbYZaW4vR0VFIkoSBeBzGj3wEiERg1+ux8JWvINLamhL9q62t3WXjk259E3jqKdXWQcVsm759lcEA+1NPIXL2LPd4m5qaAqUUFRUVqK6uTrlDPAiUFoRdKXFZl/jNmzdBCEmZJV4K/zw5pfSZ1Gg0aGlpweOPP467774bf/zHf4znn38ely9fVhSTy8vLaGl5fRpic3MzLl26lO0pfh3A07KfKYAfEUIogC9RSv+2JCfy5mGdENJIKV1dXV0tysLGZDKVXEwW+jln36FwOIxXX30VlFJec51PhO6gO8qVtmfjYJkJOkvJp0/QiUajhyrDVAi5OrPpW94CJANBxGDA9u23Y3NqCuvr6ymm6NlGJbq/9S0En34adR/8YNYJe+nkEoU6nY7bsYVCIaysrODKlSt89G1FRUXeThkF1/5nCBjIm2BZVHVpaQljY2Ooq6tL0Ul7yZ6GS6LRKFZWVnbVIRYCq7VcWlrCqVOnFFNvamx2WMPOZa0WLX//94m52CrEpxLM+7G3txe1tbVc9KiBLTDy+kilDsxcPmuhUAiDg4NobGyEz+eDVmY7AwDlV6+i9j3v4bOgfT4f/7ABr8/jtr34YsbpOLnINlmn0O2l8+dRV1fH7Z1mZmYQDocxPT2NUChUcNSvFNFNNQtCti5xdvFQMx8+n2MqRRlA+ueXEIKTJ0/i1KlTqrdh2ylBCHkrEmJS3sJ+F6V0hRBSD+BZQsg4pfSF/I/+TctTAD4K4NGvfvWrePDBBwveUanFZD4+kWx9YiIrFotxe7GTJ0/m1W2bfgwHLSYlSYLT6YTL5cL29javFy/EBP1WQv66qZ2brampQV9tLQwGQ4opetezz2YclRg9dw7rjY2wHT+e9/GpvSaYTCZ0dnaio6MDV69ehcfjwcLCAqxWK5qamlTVzRe7TufavzyqGo/Hsbm5iampKezs7GB2dhZ2u33PGir3TExGIhFcuXKFd4sWIySZhRAhRFGUsi+rmjfJ6/XC4/Hg7NmzqB0YQK5lTmlBpZTymjK5DVE+CycALC0tpfhHZnr+TAsZNzJPjopcWlqCdM89/C6P6vUIDgyA7VmeWmA2PszXarW2FmfYGCq9Pi+7GrWTdYrZXq/Xo7y8HHa7fVfUT16LuR8F6YUsCOkp8egLLyD6z/+M7dZWXKa0qJS45tIlNH7rWzA++CBQpM2Q/Nwopaouos3NzVhcXOQ/Ly0tKZayEEJOAvg7AO+mlDrZ7ymlK8n/bxBCvgNgAIAQkwoQQr6JRLONjRCyBOCPATwK4AlCyK//wi/8Av7lX/6l4P3LTcv3Os1NaWKcH6t7ZOndmpoa9PX1cXeCK1euFCwkgYMRk3JhvLm5CZfLxTNCt1rdI6PYUqNsndnyQBAdH4fh1VdRMziI6nvvRfzOOxM2Oa2t6MswKrFUvo9qIIRAp9Ohq6sLJpMJTqeTTyVjU5Iy1czvZ+2/VquF3W6H3W7Hz3/+c+j1eoyNjeHcub3pb9wzMWk0GnH27Fl4vV643e6C98O6o9Onx8hR03FNKcX8/Dw2NjZQVlbGp7nkIn0xZE1EOp0upbZR6bGZiMfj8Pl8fLRirrmmSgsZ64g/c+ZMygeXXrjAbWdWjx5F+PbbkcnEJmUed28v3B0diP34x1jt6cG2VouamRnEYrHcNaMFTtbJZ/sUuwiFqJ/L5cLS0hJvGGELd7q/aSlSRsUuqtrLl1H5oQ8BkQjqdDoEv/tduOrrs6bEM8FKBNrCYeBrX8u7xCAdpXR5rnPt7+/H1NQU5ubm0NTUhMcffxzf+MY30vfRCuDbAD5CKZ2U/b4cgIZS6kv++50A/jTb8xFC7gLwUoY/f5FS+qmsB3wLQyn9cIY/vY09pJj9l7pmMv0GWz6KkY3zU0rvlpJSeF3mWjfkVkQulwvBYJA3zdTW1qKhoSFv8261z70XSJKEWCzGvZINBgPKyspU29GwdHZZSwtI8sZSbV+D+fp11H7ykyDRKLQGA/D002i4cAENH/0oQp2dfFRi1GRC0+Ymv5bvZw0j0xusU72uro5b+IyMjIBSyi2Q5MEBSZKKujEqFEIIWlpa0NTUtGc3Mnua5jYajbusgdRCKcXq6irm5+dx4sSJrPYkuUY2RqNRjIyMwGQyob+/HxcvXlR9HPKFyO/3Y2hoKKOwVbNosfpIg8GArq6unB/k9MVEkiSMj48jEolkLB2IDwxAOn8ewaUlkDzS7sb77oPxvvvQjYRodrvdWF9fx9DQEBfgmTqVs03WUSO+ck3mybYPg8HA78AoTR0zGIvFUhp5SkGxqYr0CTj6n/8cNRcu8IuNPGLMUuKZusT5vpKdjIVM05EjF5NqRbNOp8PnPvc53H///YjH4/j4xz+O48ePgxDyyeR+vgjg/wFQC+BvkvtkFkANAL6T/J0OwDcopc/keMpxAHek/e4RAO8G8ITKUxUoYDab+c1/KcSkJElwu93Y3NyEz+eD0WiE1WpFW1tbzlGMcoq5gduryCQrXXG73fB6vXwwQGdnJ8rKyvjxFjtBZz9QagJiZQasjn9iYgJra2uIRqO8Y14JeTr7iE6H7X/9V+Dtb1fd11Bx9SpINKoYwWSjEvsoTYxK/PrXMdLTA2lgYN8ik5m2YxY+zc3NCAaDKbO3We3nfvg9Kh2rnFvSGogQkrcJOZBYxEZGRhCPx/lknGxkew5mjp7esJPPsUiSxBtBMo1FZI/Ntmg4nU6Mj4/j+PHjiZS0ioVavpBFIhEMDg6itrYWx44dU1U/USg6nQ51dXVYXV3FkSNHeGEv61RmAs1isRyqlA0hqWMG4/E4Ny5mht2Li4uwWq0pC34+FBuZzDUBJyVinKVL3GKxQJPclxQOgxRQYpCOXEyGQiHVFkcPPPAAHnjggZTfJUUk+/cnAHwifTtK6SyAzAWZCiRT5DxNTgj5UwD3A3g/pfSn+exLkEqxNZPRaDSl4zoWi6GyshItLS0F1wYe9Hxvtgaz+d2s7pFlSBobG9HT05O1m7wY9kqIsoABm9vNxDCLEl+9epXXrBsMBjQ2NqK6uhqVlZWYnp5GOBxG5/o66sfGQN76Vi4ONWmzrw0vvwy8/e2Jc1HR1+A7exaNObqzNZcuwfLLvwxLJIJGgwFz/+f/YKa+HhcvXsxrNngx5uPZtjObzby+Ut5prdVqef/CYfcHzZcDm4CTCb/fj52dHdTX16Ojo0PVi5DJC5LN/FbTRZ4JkjQNj8ViOC9JMH7pSxnvqjItWko1lsvLy6oWCbaQeb1eDA8PZ5w5nmnbUtlyMNPrlpYWxOPxFF9CvV7Px0DtVaqq0C8EM6W12WyglOLSpUvQaDSYnZ3lk2Bqa2tRU1OjOv1Q7N0lS+vHf/ITLHV1oSVLJDFnl7hOh6YvfhF4/nk0fvjDoEVO05GLyZ2dnQPvnM8FIeRRAL8N4L2U0ucO+nhudfIVk+kCixACi8XCZ0GvrKxAp9MVlRU4qAYaluXY3NzE9vY2tra2+I1ce3u76trmYo6/lGspu46w6COlibnZ9fX1Wa2V5NtrNBo+4Sz24oswf/SjQCQC+uijcD3xBCrf+c6UdDbV6RC5807kk9gNnDoF97e+hapr17J7S8rqL2tHRrDz4Q+jvb1d0RQ907kVel1Rew2Qd1pLkoShoSGsr69jZWUFdrsdjY2Ne+5JvF/R0D0Xk/mkudfW1jA7O4uqqirY7fa8bHbkgjUej2N8fByxWExVZDMTkUgEa2trKC8vx7loFIakfU2mGaEajQbRaJT/TC5eBJ5/HjPNzQidPp1SH5ltoZbbJ2jOnUMoFMLIyAhuv/32vOckF3tXq/QeaLVaLh4B7BrVtxf+iqW4u6KUQqvVcqsHZg7sdDp5AwlLJ2frzCvFsUjnz8N37BgCq6t5bZdeLxqNRuFuacGEzYYFjQbm4WEeNS5ECKaLycM8SpEQ8r8A/CaAByilzx/w4bwhMBqNWRtwJEnidfButxvxeJxPT1ISWKWo+dtPMcnWMrfbDb/fj8rKSn6zfPTo0YKP4aCQJAnLy8twuVwIBAJ8xGGueeRqML7ySiIdLUmgsRgiP/oRXq6ogN1uR+uTT8J08SKmm5tROzCQ134ppYj19yOejGZm9JaU1V+GL1xIlGopmKKzbnDWOZ/+XPuVndJoNDCZTGhpaUFVVRXW19cxPDwMQgj3r9wLP+Jiovr5cCjS3Mz0OxgMor+/Hzdu3Mgrmil/jkAggKGhITQ2NqK1tbXgiz6LBDJfM+0//VPOCTvyxZfXjYTD6DUYEHvmGVDZ3UGmBS7dPmHhS19C2GrFW97yFlWRsxQrhn0Ko7M7QCbQWKTC/YMfwDo0BENfH3ZaW1FTU3OgxuPpxdbySTBAaq0im2jD6kTl3falutOLx+NF74fNQr958ybOnTvHL4bMQoldPNRGXuViMhAI5H3zsl8QQv43EnY476KUZmrGEeRJemSSNQsy8RgOh1FVVYWamhpVguQwGJ9nE5PpU3RYTWd7ezuv6XQ6nUU1kRYrhvPZVj7Nant7G6FQCPF4HB0dHXnVqKohXdDVffCDsJw7h7W1NVwzGqH9hV9IDJ9ITpzLVSvJKMROKNzZCeLxvL6dzLlEkiRukZM+G3yvI5NK56bRaKDX61PqK1dWVrJOsinm81OK64waDjzNHQwGMTQ0hIaGBvT29hZUZ8kev7m5icnJSRw/fjynh1+2DxFLj99+++1wOp2Ix+OqOtHkNZOhZ56BLhyGRpKAaHSX+MyUgk6fzFN+9SrK3v/+gjrADqITkEXOaicnUfb7vw9EImjV6TBaVoapzs6Mpum79pM2UadUkcls+0ivVfT7/XC5XBgfH0c0GuU3FvF4vGR+laX8khOyewY3S2stLCyAEJIyS1zpueVi0u/3HzoxSRIv/N8A+DCAd1JK1XfTCXJiNBrh9Xrx7W9/Gy0tLQiHw/B6vaipqUFPT0/eKblSlNqUMjKZcrPrdvPvRLYpOqX4ru9Vmpul4pmHZTwe57PIu7q6cO3atV2DMEqFUkONDuAiaWdnB9evX8fNxx9H3R/8AR+XqJTVS4c7d6i1E1pfz/haaTQavq6nzwYPh8P7WoKgVKNpNpvR1dWFzs5OeL1eHk212Ww8mlrsGEZNWiBrL9hzMZntwFlTS7phd75iUl7X2N/fr/puOT30m94prdPpuHmumk40Qgji8Tjm5uYQam3FKaMxo/jM1KzDRKsUDgMGA6wf+ACQpxG6nFJZ4eRLesdy9+Ii2h96KKNpujytrDRRh6q0cpJTjCAlhKCiogIVFRVoX12F5oUXsH377VgFsLWVGCGt0+lgtVoLvuMvledYpveHjZuzWCzo7OzkUZi1tTVMTk7ygnvWJc4+v+yYDmma+68B/AaAzwAAIUT+RRyjlHoP5KgOCELIPAAfgDgSHfIF7efll1/Gl770Jbz88sswGo04fvw4+vv7EYvF0NHRUfDxlWLSUzGRSUoTY2bdbje2t7cRjUZRXV2NmpoatLW1qUorlkLMlpJwOMzrHtl31Gq14vjx47us0PaabA01FRUVqKqqwrHNzZRxid6nnoLp7NldwQRuJ9TUBNLWBkC9nZDatT09DX7t2jVcvXoV9fX1imnwUpNtzU+vr9za2sLs7CwCgUBRU63eEDWTmd5cSZIwPT0Nn8+nKP7yEZORSASrq6uoqKjA2bNnVX2glMQkmyRTX1+f0iktXwzVDI1fX19HbW0tej/2MUT7+jKKz0wL5EZXF9Yeewy96+swvOMdwIULwMsv73pcJhYWFrCwsICqqirodLqiazAKXQiVOpblqYd00/Tx8XG+KDY9//yuiTh48MG8jkVRkJ4+nX+di2w/doMBVU89harOTvj9fuh0Om5WqzbiKqdUX3K1+2Ep8fr6+hQrEHlKXJIkVFVVAUhEJtWIyWeeeQaf+cxnEI/H8YlPfAKPPPJIyt+T0cS/BvAAgACAj1FKryX/9q7k37QA/o5S+mim50nu56PJx35O4SFHAbypxGSSt1JKt5L/Lkj1WCwW/PZv/zY+85nP4LHHHsOnPvUpbudTDKXyqsxHzIVCIR553NnZgV6vh1arxdGjRzMOhyjl85dyexbZZeLR7Xbzm9jW1lZUVFQcqq5gpXR27K67UsYl7pw7h+ErV1BRUYHm5mbU1NRAc+kST2f36HTwNTYCv/ALqu2E8s1csWuR2WzG6dOn4Xa7FdPgpUZt97i80SkajWJpaQk7Ozu4cuUKHA4HGhoaVF/b3xBiUgk2y7q2thZnzpxRfGHVNu1sb29jdHQUtbW1qkYZMZhYZRd9NtKQTZJRemwuAoEAj/b09fUByC4+0xcYSinm5ubgdDpx6mMfg95gyOuqIEkSgsEgvF4vBgYGEAgEsLCwwOt9mHluPq9TMciNyMfq69HW37+ro08prex0OjHpcOBkctIB1esRveuuvBcLpRGN9Pbb8z53xVGPra3Q6/VwOBxwOBx82oXT6cTy8jIkSUqJuGb6IpeqlqWQxSI9JU4uXkTsW9/CXFsblo4cwec//3necRiNRjMK5Hg8jk9/+tN49tln0dzcjP7+frzvfe/j34Ek7wZwJPnfeQBfAHCeEKIF8HkA7wCwBOAKIeQpSumY0nPRxBemNGahghTY+zU5OYlIJALgcNQ7sn1kE2PpFjcGgyHFy5L9rRAhCex/uZC8xIY1zmxtbcFqtaKjo+NQWbHJUaxvNJsRHxhIEYSNFy7ATim2t7extLSEGzdu4Ph3v4v65DqrAaB/+WXgF34BgDo7oWIaabRabcY0eK5u8HwpZK3W6/Ww2+1wu93o7e3l9ZUVFRW8vjLbub8hGnDkUEp5/RmbZZ2JXAKOUoqFhQWsrq7i9OnTcLlceRmjswVOybIn02OzsbW1hYmJCe4ppQb5AhWPxzEyMgK9Xo+zZ8/m/WGLRCK4fv06dDodent7AQCVlZWor6/nYoHNhZZHAGuT80/3CmZE7nvttZyPlaeV8ZGPwN/Vhdhzz2H92DGsaDSIJ1PLJpNJVd2W0ojGQhacTPtJr0FJj7imp5OVDN9LGZksZrHQXLqEsgcfBCIRnNLpsP6Nb6Dzd38Xjz32GAYHB3Hu3DmcPHkS//iP/7hr28uXL6O7uxudnZ0AgIceeghPPvlkuph8EMDXkmLwIiHEQghpBNAOYDrpNQlCyOPJxyqKSYEiFMCPCCEUwJeKFT3p4xQPuhOb7UO+BrO6R9Y4AyQiq5ksbg4ysqh2e2aA7nQ6+RQvq9WKo0ePYmRkBD09PQU//16gFIFUqm/E/fcDSBWE5OJFaF94AdZ770XNhQuJmwGnE9IXvgBCacJO6I478hInperKzrcbvJDnK2biTllZGbq7u9HV1QWPx8Ov6TabDU1NTYrH94aKTGo0GkxNTcHj8eDcuXM56zo0Gk1GcZg+zlCr1cLj8fC7aTVotVpEo1FMTk7CYDBkHWmYa7bs/Pw8Njc3cS5p4bO9va3qGNh+g8Egrl+/nnVcZDZ8Ph+GhobQ09OD+fl5xWNMT2+ygu2RkREeRcsWtSz2YlDI9uTOO6G/8040A2iiFGNjY6CUYmJiIqUZJpNputKIRhoM5r3gKO1HWlrK+uVUSifLDd/ljTylGK1VbIRTHn0FAPOlS6h4xzvQ2tqKu+66C7/2a7+GtbU1xW2Xl5fR0tLCf25ubsalS5fSH9YEYFH281Lyd0q/L84o883HXZTSFUJIPYBnX3jhBdyboa5MDcWalqdTqjR3IBDgZujMuL+mpgYtLS05v0OHUUwqNQIxy570dXi/myhzkanDWqm+MV2sKW2ru3ABde97H+I//CGizz2HUZsNHr0e9ePjaG5uVlVqU2qLn1zd4IW+J4WapCs10bB6eHZ8rFyJ+Vey4Fj69eGWbMAhhCASicDv96O6uhrnzp2D5tKlnPUPOp2OL2hy2Jzu1tZWNDU18d/n27ATj8cxNDSEzs7OlP0okWkxjMViGBkZgdFoTJyXRoNIJKJ64SSEYGdnBzdv3tzVgKSW9fV1zMzMcP/Jmzdvpnw5Mn1J2ISY9vZ2HkXLFLUs1QevmP0QQqDT6dDQ0IDq6updpuksrZVump4+orFgGwiF/agVb/J0MjMdZ8e+vr4OrVaLcDiM2traguuflOZp54M8+irpdIjedRe0eN0aiPmgKaFob7X7HJROimb5vUAllNKV5P83CCHfuXz58slixCTrcAUOVkzKm0xcLhf8fj/sdnvBHeUHKSYBpEyyYuMKmR9vrkagg7RTk8Neg0wd1or1jdevp+wjV3e27sIFSK+9hjNHjsDv92NiYgKxWAwOhwONjY178jrl2k6pG9zv9+P69et5p8ELjRJm205+fNFoFKurqxgcHIRWq+XC95aPTG5vb2NwcBDl5eVoa2tLKbLNZhGgJA6ZobnSnO58xOTGxgZcLheOHj2aU0gCyothIBDA4ODgLlGbz8LJ7rLPnz8P8/Xrqj24gOTc8m9/G+SFF3D+gx+ENmnfkv6lULMI5opaMp+5QkcP7gWZTNPZnVkm0/RS2AsBhd9dAonPCKun1Gq1MBqNIIRgYWGBd2aylLjaEoRSTeTRvfQSJh0O1CeFs5pu7ubmZm74DgBLS0twOBzpD1sC0CLfDMAKAEOG3wtUQAgpB6ChlPqS/37nbbfdVtQ+jUZjScWkWiEm90f0eDwpTSYGg4F/J/byGLJRyPbsRn1jYwPhcBiVlZV8XOFhnyzF0Fy6BPzsZ5De+lagrS1RcpClwzo9ne3453+G7sEHgfvuA6C+O1sukEKhEFZWVnDp0iVUVVWhubkZFovlQKK3LA2+vLyMzs7OvNPgxaa5c6HX69Ha2orW1lYEAgGsrKxgaWkJBoOBRzL3ij0VkwaDAWfOnMH09DRisVjWuxI5cnHIDM1DoRD6+/sVUxpqFj1KKaanp+HxeFJCwLlI3zfzslSa0a2mxkiSJIyNjSEQCKClpQXm69dVCWxGPB7H3De+gWMPPwwSiwFf+1rKNumCKf140u1y5ChFLYeGhrC5uYmbN2/uW62lEtmEYCbT9Js3b0Kr1fJjLqWYLJWlj9FoRG1tLZ/Xmi7mmS9kdXV11kaeYgusWfTVMziIRpnPZC4x2d/fj6mpKczNzaGpqQmPP/44vvGNb6Q/7CkADydrIs8D8FBKVwkhmwCOEEI6ACwDeAjAvy/qRN5cNAD4TvIzrQPwjXe96113FLNDefPjXkYm5R6obrcblNKUMYzyz/Pm5mZRx1EK03M1sIla8nGFrAmvvLxcVfDiMMDWUPP16zD94R9CE41C+ou/gPcv/xKau+6CdP58bpu8ZDq7ORwG/uEfMpqNK22bvk6bTCY+59rtdmNxcRE3btyA3W5HU1MTjEZjydZ2teRKg2fqBt+rWeBKsPpKo9GInZ0drK2twev17tkkpz0Vk+Xl5YhGo1wcqr0rYY+X2/UwQ/Nsj89EJBLB0NAQqqurcfbsWUxPT6teXOTNOqzbOpOXZXqheDrhcBjXr19HQ0MDrFYrQqGQaoENJDrhr1+/jtvGx7lvl3yb9Dvw9NdLyS4nXVDK0ev1MJlMvCtyZ2cHwZ/8BJ4XXoD75Eno7rln3zrE1X6Z0scNsnTZ1ne/C9PFi8DJk1jP08Kn0GPJhVIdjFzMsy5V5sfKhKfcF1JpP8WQblqeS0zqdDp87nOfw/333494PI6Pf/zjOH78OL74xS/iU5/61CcppV8E8AMkbIGmkbAG+o8AQCmNEUIeBvBDJOx+vkIpHS3JibwJSDYunUr79Z8Vs0/557qUYlKe5nW73SnTmZqbm7PenB50mjrb9iwrkj6usLm5ma8vS0tLRa0X+xF1C4VC3PicjcS1jYxAE4uBSBI0sRjs4+MYP3YMFy9eTDhZ/O7vZlxD+XVNkrKajSuRrY6RZXVisRhWV1dx/fp16PV6GAyGoua/F0M2U/T0NPhepLlzQSlFZWUlmpqa9vSztC8NOOxuV61nlFarRSAQwKuvvqpo16P0+Exiko1FPHLkCDf+zGeRZM1Ag4ODMBqNWbuts+3X4/FgZGQER48ehc1mw9raGiilqgU2s0Hq6+tDJQD8zd/s2kZJTMp/VrK5iWQRk2wf7P/VY2No/E//CYhE0GEwYPmrX8VKWxvGx8dRXl7OU8/7HbXMhtFoRNPCAsp+53eAZE3gXG0thpJNIyxqWVlZmbt2JhnVNbW3Q/PWtxZ9bLkWCJ1Oh7q6OtTV1QEAb+RJT+fHYrE9EZOBQEBV9+IDDzyABx54IOV3n/zkJ/HJT37yiwC39Pm00raU0h8gITYFhwT2PShWTIbDYWxtbcHr9eLy5csoKyuD1WrFkSNH8qp7PExiMn1cIUvBd3Z2ZiwF2m9rITXE43F4PB44nU5sb2/DOjGBxslJ1L3tbdDfey8IIZg4fRp4/HF+nYnfcw+amprQ2NiIlZUVXEl6Rba0tOxKO/PrWnL4RqbrWqHodDq0tLSgpaUFPp8PY2Nj2NraQiAQQFNT054OW8j2XubqBgcKn5xT6Bofj8dTyr1uyQYchlzsqbkrWV5ehs/nw1133aUqHa3VahUXveXlZSwsLPAGFaXjyQXr0O7r61OqBUsh0+K7srKCmzdv4vTp07xWht+xqxDYkUgEN27cwJkzZ2A2mzNuwxatTGluJZubfEgXo3Wjo6h+17tydoiXYiEtJhooP24NgJaZGTS8//3cNH1paQk+n4+n8a1W6y7HAXlU94hOh41vfhN4+9sLOh4mSs0tLdC8+92qtzObzXxUmTydv7GxASDxGqkVxtlg2x7GcYq3EoSQuwBkmhv+RUrpp/bzeNRCKeW1XfmISSa02LQZnU6HqqoqfhNeTJ3xXs3mzgXzfPT7/bh27dqucYWH1fNRCRYZdjqdiEQiqK6uRm1tLY5sbaHi934PiESAL32JZ6y8x4/zWmry1rci1NQEQikMBgMXTNvb2zztzHx3DQYDv0atPv44rB/4AAwqegEY+a71zAZPr9dDp9Pxpp2mpibY7faiB3cUcnyZ0uB+vx8LCwt5m6KXapziXmYQ92UCjlarVeUDGY1GMTw8DJPJhMrKyrzqGuXiUJIk3Lhxg49XTP8wqV2cNjc3MTExgbKyspxCUmm/zMomGAzuOg75ApdJYFNKMTk5iVgshrvvvju1mURhm1xpbiWbm3zIJEaV0rPyyTaRSATr6+uoq6s7kKil0jQeYLdp+s7ODlwuF0ZHR3fVKxoU7HMKEZNyUXpUp8P2t74FvOUt+e9Hls43mUyIxWIwGo1cGLNIsZIwVsvOzs6ejxd7gzMOIL2G8REkTNyf2P/DyU0+aW55jaDb7YYkSdz2qrOzk6/7Ho+n6Jub/YxMpo8rLCsrg0ajKXhcYa7yp70iHo8jFothcnIS29vbfHxqeke84eWXM2asWC21wWCA6amnYLp4EeR97wNNllWxNSgajWJlZYWnd1taWmA9fx7Lej0sx4/nddyFBA6Y+bjdbofdbkcoFMLy8jIuXbqE6upqNDU17YqeFkq+wk6eBn/ppZcgSVLepuilnM29V+xbmjtXJNDr9WJkZASdnZ2oq6vDlStXVO9fHmlkdZYNDQ1oS3afKT0+Go1m3B+lFLOzs3C5XDh79iwGBwdVHYd80YpGoxgcHITFYsHtCpNXci3UrPmluroaZrNZ9QzZTMfDSLe5UQPbh1oxqtPpUjrEr1y5gkgkotrXMtMxFLoQsOOO/OhH2Ojrg0PhuOWCuK2tbVe9Yp3DgduS9UGSTofInXeikHkaKVFSSmF45ZWCxKSceDwOo9HIF1IWTXE6nRgbG0MsFksRxmqjKeFwuOCpIQKAUuoE4GQ/E0L+FMD9AN5PKf3pgR2YStK/b8wzVW5vwzqUm5qaMtaRF5uZ2OvIpNxqjEVU5eMKo9EoxsbGipp7XawYVvscgUAATqcTTqcTsVgMsVgMdXV16O7uzigo1GSsyMWLcPzar4FEIsAXvrCrUVSv16OtrQ2tra3weDxYWlrC+Pg4JElCJBLJex0pREymN+10dXWhs7MTLpeLR08bGxvhcDiKfi+LibIXYoouxGQSrVab1VR8aWkJi4uLOHXqFMrLyxO1hHksHmyxyTYWUenxSsRiMQwPD8NsNvPUTD7ekUAiojM0NISuri40NDRkfGymBcbv92NwcBCdnZ2w2+3Y2NhQHVqPx+PY3Nzkqc5STJ+Qk68YJYRAq9WitbUVHR0du6KWamsti216kc6fh6e7G0GVpvLyekVKKQLHj+NmRQXws59h5cgRaOvrYXc6M5qmZyLd0zF+990oNlGm1MjDpgm1tbUhHo/D7XZja2sL09PTKVYr2Syf9suf7M0AIeRRAL8N4L2U0ucO+niyIS+XkSQJa2trcLvd8Pl8MJvNqKmpQXd3d0oTWCZKZS9USjHJbraYKI5GozzlyyKq2bYv5Pn3CnbT63Q64fV6YTabUVtbi76+PhiNRly5ciWnh7GaIIHmhRdAMjTUyCHkdTPtWCyGixcvYmRkhJfo2Gy2nK9HIa91tqYddn1hHoyvvfYaDAYDmpubC36uYtfFfLvB3/Rikr25mWZtx+Nx3LhxA5IkYWBggH+JC/nyhcNhTE1NYSAeh+nv/i5ng4/S4sREXHt7u6q0thIsoqjkhykn0wLldDoxPj6OEydOoKqqKuWxar6E4+PjMJvN8P7whyi/cgU4fRrupNn3YRAG6VHLfKfxFEOhgpQQgvLycpS/973Ae9+LtZERWCyWnKbpSuyaWX7+fNFiMpc1kFarhc1mg81mA/B6B+rs7CyCwSAqKytRW1uL6upqfuyFLLIulwu/8iu/gvn5ebS3t+O5556roZS65Y8hhLQA+BoAOwAJwN9SSv86+bf/CuA3AGwmH/6HyQadWxpCyP8C8JsAHqCUPn/Ah5OVQCAArVaLhx9+GO95z3tQXl6OYDDIoyaFzIAvllKIuVgshrW1NbhcrpRxhb29vaqiZgdtei4/DrZmulwuXo7DDN0LFhyyIAGr6a6qrQX6+xN/v/deaAwGIBpV3VCj0+lgMplw4sQJhMNhLC4uYnJyklv6ZHrdC01z59pG7sHo8/mwtLTEjdGbm5tV14cX4zOshJpucEmSCnYfeUPN5lZqeGHG301NTWhpaSn4zWGTaCRJwnlJgiE5YzibZ6O8xpLNGHWeOIGx6mrcdtttXMTlA6UUMzMziEQiuOOOO3LWBirVVy4sLGBtbW3XyEk1/pWBQACbm5tob29Hy9ISL6iWvvENTJaXY6qtDWVlZVmjgNk8KPeCXLWW8qhlKex4SmXpw/zjmG+cWtN0Blu4vdeulWw2dz77Sffl9Pl8cDqdWFhY4DdUW1tbIITk9Xo9+uijeNvb3oZHHnkEjz76KJ577rlHAPxB2sNiAH6PUnqNEFIJ4FVCyLOUUjaP+68opf9T9ZMecggh/xvARwG8i1KaqRmnmP2/C8BfI2Gt9HeFipZwOIx/9+/+HZ95fccdd+Dee+/F2NgYOjo6SnnIeVNIdFPeoMZse4LBIJqbm/NuUNtr27NcSJKE9fV1PrebDTa47bbbSl6DLq/pPq3TIXTsGPeHXP7qV2G+dAlVyZpJNbA1t6qqCsePH+eifnBwEHq9nkcrleap50u+TTvHjh2Dy+VCdXU1D2o1NTWhoaEha1nZXmZsMnWDs+EihSAfp3jLNuAAr6c55WKS1aEpGX/nA0snt7e3J+6oX3xRlWcjW5z4nNBwGDa9HgPf+x50BQhJeWq8rKxM1RdcfrfKjMwppYpzwnPd2bL0PptVrXv88ZS6vPb5eTT+4i9yaxmlKKD28uWMHpTF3lWr3T49aun3+7G1tYWRkRHs7OzwLrhCo5alFJPy90itaTobTcgo1R1uMablGo0G1dXVqK6uhsPhwPj4OCil+MpXvoKbN2/igx/8IO6//3784i/+Yk6LrieffBLPP/88AOCjH/0oPvvZz74faWKSUroKYDX5bx8h5AYSM7rH8AaCJN7YvwHwYQDvpJRe3IPn0AL4PIB3IDFl6MrY2Bj6+vry3pfRaMQ3v/lN1NXV4Z3vfCfe8Y537Km9Sj6oiezJ6zmdTief3221WtHS0oKhoaGCRfF+NwBRSvkNHvN9DAQCBQnhfJHXdBNKoXvpJUST19DQ6dMInzmDypaWHHvJsn+djjtSsOjg5OQkGhoa0NzcnHArKeC1LrQMghDCa82DwWBK005zc3NKtkb+XHt9g5GeBh8eHsba2hrW19fR2NiIxsbGfZuQppb99ZmkFFNTU/D5fBmNv+Vku/ivr69jenqap4Pn5+fzNkXH888D4TA3ZTW+8gri99yT17mxCGtbWxscDgdefvllVdsxQRuJRHD9+nXU19dnbBjKVjO0tLSEpaUlnDlzBisrK5iYmEDb0aNo1+sTX0qDAdG77oIkSQnPxWQkmNXRsSjg0W9/G2XJ10Le0af2S5Mrqpnvl09e99fe3o7r16+joqIi71pLOaUSk9kWk0ym6fPz8/D7/aiqqkJtbS1qampKdodbqsWC+ZHddttt+NKXvoRf/MVfxJ/8yZ/ghz/8IZxOZ04xyRY6AOz/WW+lCSHtAE4DuCT79cOEkF8DcBWJCKZbadtbgL9GImX/GQAghMjvascopd4SPMcAgGmaMC8HIeTxJ5988mQhYhIA9zM1mUx8pOJBR+WAzJFJNq7Q5XLB6/XybuUjR46kjCtkVkeFsh9iMhKJcPHIXBRqa2vR3NyM1157rajocKZ1T2nNzuR8sRew6GA8Hsf6+jqGh4eh0WgQiUQKer3zXQPTn8NsNqO7uxtdXV08GOD3+1Msj9h2hQYzCkGj0XBHmaqqKp4GN5lMaG5uztkN/oYSk6x7+urVq6ipqcGZM2dUNZMovWlyQTowMJBSR6BmzBMA/oEdrq3F6TzrQOSw+sZCIqyEEEQiEVy5cgU9PT18Ic/02PQPIrMdCoVCOHfuHCilaG1tRWNjI2YqK7H2Z3+G+rExxO6+G+beXliSkStKKY8SWywWPt86GgiAfuUroNEoqE6Hpe5umH0+VV+AfCfrZNpHLjFqs9m4iz/rVs6n1nKvIpPZMBqN/E6SUgqv14vQT3+K7RdegOHoUczZbEV7Q5ZinCKQWlvD5nL39fWlRLre/va3Y21tbde2f/7nf57XcxFCKgD8K4DfkQmrLwD4bwBo8v//C8DH8z+TgyUZlfwoEqnnzyk85CiAUojJJgCLsp+XlpeXi96pfD43ULrvTaGw9U9eksFGMFqt1pz1gsUeeylqHne5aiTHSbJz0Wg0qK2t5R3ke/16Z1qz5TXdw7W1OHr+PIo5EjWfHa1WywXbzs4OLl++jIsXL6Kurg4tLS2q5pjvZdMOszxi4s1oNO57ZowFMHKZoiv1abwhfCaBxMHv7Oxge3sbp0+f5k0AuWB2QvIFQj4WMV2Q8tR1DlN0APxu9vxDDyF24kRO8ZkOpRQ3b97E+vr6rvpGtbhcLng8Htxxxx05C3/TFzM2kaeqqgq33XZbyod0cXERkUgEt/3Gb0Cj0cDj8WBrawuzs7PQ6/WwJcWLyWRKjLhM3vFr774bO08+CcPLLyN8xx2QurqwsLAAp9OJaDSKxsbGjDWAhUzWkZOvGE3vVlbbIb4fkclsEEJQMz6Ost/8TSASQZtOh9W2NiwlC8ILnX1eqjvPWCyWc5Tic89lbkRuaGjA6uoqGhsbsbq6CgAbSo8jhOiREJL/RCn9Nvs9pXRd9pj/A+B7hZ3JwUITX9b9mO2260NYis+3yWRCKBQC8Pq6elDm3Kw0x+l0Ymtri48rbGlpKbghIV9KIUaBhG0dS8MHg0GepWhra8tao7cXAiDbms1ruvOw5ysVFRUVMJlMOH/+PDY2NjA6mpiu2tzcjIaGhozr3F427bS1taGtrQ1erxdLS0vY2triU/rUCF1GKTuys3WDp6fB2fe3VE1gmdhzMTk/P4+VlRWYzWbVQhJ4PRXNFgw2jlA+FlHp8dkWPdYkw4puKysrc07kkW9LktY7o6Oj0Gg0ivWN8sdm2s/s7CycTieqq6tVdZDJxaQ8rd7Q0MCFjSRJGB0dhclkwqlTp/jzy1OuwWAQW1tbmJycRDgcRk1NDerq6viUGun8eQSS3XtWSlFXV4fJyUlUVVXB6/XyGkAm1JitTKkn6yiJ0WyvqbzWUnPpEuLf/S42+/ow0tKSErUsVa1LMenp9Jqk9ClCLpeLR1tramp4tDXjCM9Ll+B44gkYfumXgDvvLOa0UkRDIdNv3ve+9+GrX/0qHnnkEXz1q18FgCfTH5OM2n0ZwA1K6V+m/a0xWVMJAB8AMJL/WbypWAIgL2BrLtSFQo48MsnWlv0Sk+njCo1GI8xmM6qqqtDX13co0u5qkSQJ29vbWF1dhdfrhcfj4Ybu2Sy5So328uWs6exMazar3XS5XDCbzSnXY7UUegOv1Wq5KAoEAlhaWsLMzAxsNhuam5t33ejulZiUwz6DLpcLMzMzGB0dBaWUC91c35G9svfJ1A3OIqnycbu3dGSysrISAwMDuHgxv/pzNhMbeN2HUj6OUOnxuUzAh4eHUVFRgVOnTmFoaCivY5EkiRuRNzY2ZuxAz5SeBxLpSDbh5+TJkxgeHlb1/GyfbrcbrMC+qqqKR25ZxNbhcPAOYyXYZAJ5zeTGxgaf8mOz2WC1WmEwGBCPxxGPxxEKhfgHtaOjA9FoFE6nk9vKWCwW1HZ3o/bf/g2Gl18u6WQdOWq++PIIZ6XBAPtTTyFy9iyPWjqdThiNRi6IC+2ELCbCmakmSd7ZLjdNX19fx+TkJMxmMxfFzFKDnW97OAzy9a8XVF4gR34zVoiYfOSRR/ChD30IX/7yl9Ha2goAjybPzYFEp/EDAO4C8BEAw4SQ68lNmQXQ/0sIuR2JNPc8EnY6gsxcAXCEENIBYBnAQ+973/uK3qk8MlmqiEam74xcsDCrG4vFApvNxo22PR4P1tbWbgkhySKpzNTdYrGgoqICZWVl6O7u3vfjqRodRdkf/EHWdHaKyExmeZxOJwKBABYWFmCz2RAKhbC4uIiKigp+TvtFWVkZenp60N3djc3NTW6GLhdxhYrJQsVdRUUFjh07hkAggOXlZVy8eBEWiwXNzc0ZS62KTXOrOValNPjOzg4mJyfhcDhgsVgKen417LmYtNlsqudgy9HpdIhGoxgZGUE8Hk/xoVQi27xt1vXNTMDl6V01aDQauN1uTExMqDZET3/jQ6EQrl+/zjvZotGo6kWaEMI7uc6cOQO9Xs+fw+fzYXR0FEePHs3ZICFH7j1Ik5MTNjc3MTY2xufPejweVFVVobq6mtdaajQa1NXV8RGErO5nVqeD4e1vT0Qtg8GUcV25zrPYMY8MpQindP48j1ouLi4iGo0WPY2nGJRqkhTPJd00PTlXd3x8HNFoFBaLBR0/+hHKIhFo0pqmCkUuJlnNZD7U1tbixz/+sfxXLgCglK4AeCD575egkJ5N/u0j+R/1mxdKaYwQ8jCAHyJRn/mV48ePnyx2v+mRyULWbznpN9isKc3pdMLv9/MpOpmsbkphfL5XsOk5TqeTR1Jra2vR3d3NAx/sb4Wi9jqhVHdeMziYNZ0dHhhI1J/fvAmn08nXRIfDAZ/Ph+PHj4MQwudeBwIBTExMIB6Po6WlJWvqmR17qdZVeQQuGAxiaWkJr7zyCmpraxGJRPY8Mqm0XVlZGY4cOYLu7m44nU7Mzc0hGAzySTsHZTwuT4O7XC7U1NRgenoaJ0+e3LPykH1pwCkESilGR0fR0tKC1tZWVUW8Sove+vo6ZmZmUkzE02d55yISiWBychJnzpxB2eBg1hpLpYVve3sbo6Oj6Ovr4ynnbB3aclj6MxqN4ty5c3xh1mg02NjYwOzsLE6dOpVX7UY6hCRNucvL0d7eDr/fj+vXr8NgMMDpdCISifCJKczwXZIkUEpRWVnJ7RPC4TCcTicmJycRiUR4g4+aL22uyTpq9qEmwmk2m9HY2JhXrWWpYYu49+pVVY+Xvz8sqry9vY2Nvj5U6XQglAJ6Pfz9/dAVsXjLxWQgEDg01jCCzCQjunJj9z8rdp9KNZPFQAjB1tYWPB5PyrjCtrY2Vc0mpWiAKRXyGztWT84iqV1dXRkDHnt9/Jnqzrdvvx1IWxNZVsrpdMLj8XD/4XQxn/6+MIHS19eXknrOp1GmVJjNZi7iWBBkZGQEbW1tsNvtqsYPF1r2pLQdaxC12WyIRCK7jMdZ0OYgptgQQtDQ0IC6ujpVr0uh7KuYVHsn4HQ6sbm5iY6ODrS1tanad7qYpJRienoaXq8X/f39KWpc7QdIkiQeCerv70fZ4GDClzKLKXr64ru8vIyFhQWcOXMmJVqnxog8FovxdLw8RUIIwfz8PJ8dXso7jZ2dHYyMjPAILEtDbW1t8WNh6XD56EtWV2W32+FwOBKL7o9/DOnLX4axvR0jyTFftbW1Rc1FzUauCGf650/J1/KwT+MBwNP0+KVfQqi5Gavf/CZ0b387li0WhK5c4aPhLBZLXouHvCZqZ2cn7zS34I2B3BpIzTqVDvsuMcHFaoHr6uoUxxXm4qDFZCwWQzQaxcTEBDweDy85UTs9Zz8yHpnqzr3Hj8P/5JOgzz+PrePHsWQ0InrtGmpqalBfX48jR46oFiny90CeemaNMoQQtLS0oK6uju9zr50ACCGor6/H2toaHA4HPB4PLl26xFPO2VxWiolMZnvNDAYDn1Pu9XqxvLyMyclJWCyWgqP8+2XvUwz70s0NqOsKpJRifn4em5ubaGpqShFfuZCLODbSsLKyUpUNkRKRSASDg4P8okwIgeaFF3KaorPFl1KKyclJaC5dwp0rKwAhKcIz1wIZDAZx/fp1tLa28kXZZDJBp9Phxo0b0Gg0uP3220v6AXM6nZiamsKJEye4kJCHyzs7OxGJRLC1tcU9uKqrq2Gz2WCxWHjENx6PQ3flCho/8hEgEkGjTgf3v/wL1qurU9LoNptNtVBT+8XPFuHM9oXMp0O8FJRqcZDOn8dCshmsFqmTP+bn57OapqeTXjMpIpNvTtLFpJrIZCQS4XWP8nGFx44dw8TEBDo6Ogq+idzvNLf8xtLlcvGyqHzFF2M/xHB6ViZ6111wOp0IhUK4ZDDA9MADiejj7CzKLl9O3GwXUMOZvn5oNBpu+r2zs4OlpSVMTU2hoaEBLUWYm+cLpRQmkwk2mw2dnZ28rj8UCqGpqQkOh2PXjXUp0tzZIITwgRDxeBzz8/PY2NjAlStX+KQdtTdWpbpe3NINOPyJklY/mV48NkXGZDLh3LlzWFhYyEvFs8hken1kIfh8PgwNDXH/x6GhoUT0TYUpOkn6R46NjcE+N4eOz3wGiESAxx5LiWRme1P9zz0H93e+gxPvfz/MyYkvbEB9IBCAxWJBd3d3ST8YS0tLWF1dxZkzZ7KmeA0GA/cEY8Jla2sLc3Nz3HrIarWi/JVXUrqWzZcvw3H33WhuboYkSSmG6Ww8GGv+AXbX/5TiDjeffWSLWvr9fszMzOTstM7GXnXI5mOanh7RTheThY7vEtzayGsmMwk5+U0L80m0Wq2KU1qKFYP7IcaYAbrT6YTX6+U3j319fTAajbhy5Qr/Tu03am+it7/1LUSfew5rR49iU6NBjcsFnU6HM2fOQKfTJVLhv/zLRfkBZ6OiogK9vb2Ix+N8ZCKbYlZXV7enQka+tstTzuFwOONUm72KTCrBehSCwSA6OzuxvLyMV155hX9nco1wFmJShlarRSwWUxQq8rGIzNoiW0NNpv07nU4+FUfJvFMNa2trvA6RRWbYYkgvXMhpis5GH3V3d6Ppxz9WNd5RjvN730PDr/4qamIx4BvfQPC734X5wgU0NDRgc3MTvb293F4oEAjwCF9NTU1B4oSVAwSDQZw5cyavfaQLl1AohK2tLUxPT8NQW4vTrJ7PYAC9996UYn426gxI1Oc5nU7e3d68uIjO//SfQKJRvuihgHMrlSBNj1pevnwZlZWVWF1dxcTEREG1lqW0KcqGkmm60+nE4mLC65pFLSsrK3fVTOaT5na5XPiVX/kVzM/Po729HU888YTixZcQMg/AByAOIEYpPZf8vRXAPwNoR6KT+0P01p1+c0tjNpvhdideer72yWoFXS5XyrhCNT6JxYjJvYhMsvIdFn0EEjZqDocDvb29Jb3o7pUYZoKeGZ/rKytR+4lPoKG2Fl3Dw9A99RRGbDZoBgYAFO8HrBatVsvHy7744ovcA7GxsRFNTU17Vuak9J4ZjUZ0dnaio6ODT7UJBAJwOByKoxLVUEytJZtmw+o9t7a2MDMzg1AoxIM0SmVrIs2N19/gTOKQibd0AajValOmMGSDUsobRdKn4qglW42lfDHL5kvJvtTHjh2D3W5XPd5R/vyVzz0HTSyW8oVf7+nhs8yZwGWRwe3tbWxubmJ6epqH+W02m6paHuaZaTabceLEiaIXUOZr1dzcjK3mZlyNx+GYnMT6sWOIlJfDtrEBq9UKo9GY0lHPtmttbU38/rvfBYlE+GjH0DPPQHr3u/M6PqWCdGqzleQiwep0MtVaqvGHLFmaO4/9yNMuQCIa43K5sP300wi+8grit90G19vfDkmS8k5zP/roo3jb296GRx55BI8++igeffRRPPbYY5ke/lZK6Vba7x4B8GNK6aOEkP+fve8ObKs813+OpiXLS7IsD0meseM4XtmEUTY0jEALBOi95ZZfW9oL3RTSS2nTpqxCe1vgtrSMQltoCwkQRhglkFIgzfZIvKck2xqWvLTX+f1hfx9HsrZkM5rnH0hi6RwdH33n+d73fZ5n+8Kfb1/80lNYapA2t8fjgcvlgk6no+uLQqFAbW1t0iNI6ZCpTJEx0oonc5xERd7Y2LikgrtUz59shnMVCmDB/5erhHc6nXQ+urKykm4EuWtfi0AAV10d2E2b0vYDTgV8Ph+rVq1CIBDAxMQEjh07Ri3q5HJ5xkh7vOvLMB+m2ni9XoyPj6Ozs5N2ycgoW6LHSodMcs+JOHaQczp8+DCkUimNSSTHSeeYy2WptaxtbuIbCcx/yP7+ftjt9kXkDUi8MknmI3k8HtRqdVJEkvyCiNBFJpNFnLGMdy4sy0Kn08FoNKKoqIiq2hKpZAIftvilUimKrr4aeOop+oWfqKvD6PBwxPYzaS2RCh9pKZw8eRJ+vx8KhYKakod/JjITWlJSArVanfA1SwQTExPQ6/VovuGGeasMzrl1d3fTcyssLEROTg4V8JBrLLrwQuC3v533YhQKYWtqgtPpxPHjx+mCEG/+L9IunN26NaUvFrfCGVjY5RNEm7WMV7XMZJ52qu1yoVCIkpERSL/5TcDrRaVAgIGCAvy/Z55Bf38/ZmdnIZFIsGnTprjfqz179mD//v0AgBtuuAFnn312LDIZCVsBnL3w/08B2I9TZDIpMAyzo7S0lEaz3n333diyZUvCr/f7/Th48CBeeukltLW1YWBgADfeeCMKCgpQXl6e8v2abmUx1ddz4xcdDgdOnDgBuVy+bJGF6YBLCFsFAgzl5cGg0dDYxcrKyqjG5+HBCIL33oNv06a0LdjS6ewIBAJoNBqo1WrMzs5Cp9Oht7d3Ue51qkjm3EQiESoqKpCfn4/BwUHo9Xp0d3fTymm8c0l17Y51juScuEk7vb29UKlU1Ds6EwR2KbGsbW5CFgiRiZXTnQiZnJubQ2dnJ6qrqxEIBBKuZAIf7pZJokxlZSVKSkqi/my0xSwYDKKrqwssy2L9+vXo7e0N+dl4CTtc/8mSkhIEqqrgfuUV8N59FyMVFZipqsKaVasSuiGIfQwhNVarFQaDAbOzs8jJyYFSqYRCoYDH40FnZydWrFiRMUEJ8KGAanp6ms7pRDs3Im6ZmZmBTCajrXqhUIjAxo2Ye+EFuugpN26E7uhRNDQ00Pk/7q48Uos/0i48lcUwvMLp2LMHTIyWXjyFOKlaMgyz7JXJiOcb9uDRDA5i9+7d+K//+i+0trbi6aefxptvvomdO3fGfB+TyUS/PyUlJTCbIyYpAvOG5G8yDMMC+B3Lsr9f+HsVSb9hWXaCYZhTA5sp4Dvf+Q5uvfXWlF7761//Gj09PXTe/O6774ZOp4NIJErrHlvOyiSxJ7PZbHA6ncjJyaGbz0wLFhNFspVJn8+H4GuvQerxgAkGwbAs5B0dKLzssojjBOHjPNGCEYD4FmxLDdIdaWxspLnXR44cgUwmg0ajSdlUO1Wim5WVhYaGhpAM7niV00xVJiMhXLRjNBrR2dkJh8OBiYmJuL6eqRwzU1j2yiSJRSTilmiIRyZJe7ypqQkymQwmkylpI3Kz2Uw9KGMNwEYjk16vF21tbSgqKkJ5eTklCImeRyAQwNGjR9HkcCD3z3+G/4wzwGzaBO/ategUiVBQUICGioqUblyBQEANXsmsnMViofMZarU6oVZ4omBZFj09PWBZFs3NzTFv4HDCZbfbYbFYYDAYAMybXxc2NyN782awwSCcDgc1TC8sLKQxkqTqQMQ/3JjHSLtwtq8v6WsZXuHk//OfYM49N6HXxqpaTk1NgWEYTExMpOVrmU5lElicyENGMTweDy6//HJ8+9vfpj97/vnnw2g0LnqPu+66K5lDns6y7PgCWfw7wzA9LMu+m/IHOIWM4Xvf+x4A4M0338TLL78MIHM+k0slwAmfGxQIBFAoFKioqAjpXuj1+o/UXijWsVmWhWf/fgT27cN4bS1mGxqgbmmBXCQCfD6wAgGyLr4YwShEMpK/JFn7ThQWombDBnwcp+1I7rVWq8X09DR0Oh16enrg9Xrh9/uTsjVLN06Rey4zMzMwGAzo6emJOOcZDAZTGqVLlthxZ0/fe+89zM3NUeFnWVlZXNFOpGN+ogU43JlJi8UCu92OdT4fpH/4Q8zWbzQySSx3HA5HSHs8GSNylmXh8XgwOjqK9evXx32QR1pQwxXfsX42EiYmJuByufAZkQg527YBXi+EIhFmdu/G8awsVFRUQKVSJfR54oHsdhwOB4RCIVavXo3Z2Vn09/fD7XaHiHhS2cWQmMjc3FxUVlYmdcMyzIcxgsR6yGq1YnR0FHa7HRKJhCYxCASCkHa4TCYLsSwiAizymRQrViCfk5+eyoITyXIj5VxuDom2Wq0wmUwhaTyJzFqGI92dZ3giT+3GjWAQ2Rrorbfeivo+KpUKExMTKCkpwcTERFQl+EIaDliWNTMM8wKADQDeBWBiFrK5GYYpARC1tHkK0fHwww/jj3/8I9atW4df/OIXKSmQU7EGioVMqLm5cLlcdG7Q7XbTDkVFRUVUAvJRelVGWnO4sYXMv/6FNbfdBp7PB5VIBOfLLyN4xRVwlZRA8N576FAosDJaUlYUUQ2pQM4dPfqxMXyPBoZhqJDT6/Xi/fffp+prrVabEGlKl0xyzyU/Px/5+fnw+/3URUUsFkOtVlPz8eVSgRPweDzq60lEOx6PB6WlpSgpKYlKbrmuIUs91rEslUlSrmVZFptYFuIrr4xp/A1EJpMkgzo/Px+tra0hFyfRGctAIEAf3g0NDQlVhMIXQ5Kq09LSskjxGq+lw7IsBgcHafJAVthiMLlrF+rvuivuF4h38CCdxYw3+0KO6XA4qGI7NzcXarWapiFYLBb09fXRjO7CwsKEVHdkZIF4eaULkUhE1ceTk5Po7e1FYWEh+vv7IRAIqPWQRCJZZJiuUqmoanl6epoqy4lwwO/3J/2FCq9w+lpbwevuTvtzEl+08vLymLOWRLAU9fwyYDFEHjyzhw/T60Ni7hLF5Zdfjqeeegrbt2/HU089ha1bty76GYZhsgHwWJadW/j/CwH8dOGfXwJwA+bzvG8AsCetD/UpBcMwbwGI5Hl2B4DfDg4O/phhGNx555343ve+hyeeeCLpY2RlZcHr9QKYX8+4s+4pnnNaZDIQCMDv96O/vx/T09MQiUSQy+WoqamBRCJJ2PPvoySTRNTm3LcPzD/+AVtTEwRnnonS0lIUWq3zostgEKzPt5gQHj4MIHJc4nKIaoiafzkgEokgFotx2mmnwWq1UtJExsBiJQxlgkxyQeY8NRoNZmdnodfr0dfXB5FIlFKhJxNiGB6PRwsSHo+Hinays7OhVqsXteZJRw9Y+hSmJSeTPp+PflipVArB7t0J2eWEk0MyH1lTUxOx6pEImSRG4MmaqXLtMYaGhjA1NRVRNATEXjhJBS8rKwutra1oa2tDt0qFxoX3CQoEKLrmGggSIJLiSy6Z968UieB59dWohDIYDOLkyZMQi8VoamqKKC7iZnQToQxRuhERT7h3HDBPOJZi9hKYr9waDIaQyjGxHiKtehJjRtTJ3Kplbm4uVegF3nsPvl27MKXRoLupCYWFhUlVALlzRkG3OyM7vPCKYrRZy5MnT8asWnIXi0zCu9A2SxTbt2/HNddcg8cffxxarRbPPfccAIBhmFIAj7EsuwWACsALC9dPAOAZlmVfX3iLewE8yzDM/wOgA3B15j7Npwcsy56fyM995StfwaWXXprSMcRicUbjFJOdmWRZFi6Xi84+kranXC5PKUEHyAyZTJawkI260WiEzWaDtKMDq7/1LTA+HypIS7qqCoEzz1wUeRiOaO3sdEU10eD1euHz+dDZ2QmXywUej4esrKxl857lekW63W6MjY3hX//6F+RyOTQazaKuyVKQSS5yc3PR0NBAxbqjo6OwWCzQaDQoLCxM7DmS4flFsViMyspKVFRUhLTmi4uLUVZWhqysrE/XzCRpqzqdTszMzCRsl0NMzoF5YjE8PEznIyOBZEZHg81mQ3d3N83Hnp6eTrgtzuPx4PF40N7eDrFYjDVr1kT9BUVbfInQhlTwAoEAmpqa4KypQQfLQnLwIOzr1kFYUgLlzEzMZJhISTyRFhFSyeWmEcSqaHJn/CoqKuDz+UJaziTtRi6Xw263o7u7G6tXr07Z0zMaRkdHYbVa0draGtK24loPEUsHUn0Ui8VQKpWLrIcEhw+j4OqrAa8XhQIBrH/7G+xVVTAajbQSm4xHZKbiwVJJ44lUtVyqxSLZloxCocC+ffsivc84gC0L/z8EoDnK8awAzkvpZE8BALAwJgAAeOGFF7B69eqU3kcikSx7mzs8Lzrchujw4cNpbVjTJZOJmly7XC5MTk7S3G6yCQSAusOHwfh8EVvS8QhhLI/ITIhqyOz65OQkbDYb/bxVVVW0+9bR0YGBgQHMzMxAq9VmfN2PhqysLFRXV6OqqgoWiwW9vb0IBAJQq9UoLi6mm5VUyGSya6dAIEBOTg7UajUkEgkMBgP6+vqgUqno38U63lK0msNb88Qwns/nJ2V5lC6WZWZSJpPR3WWidjnE5Ly3txcOhwMbNmyIOZAba2ZSp9NhfHwca9eupaKTZBZJv98PnU6HmpqauDY6kd53dnYWnZ2dWLlyJfLz82m5OxgMYmBgANLNm6H+4hcRCASoofTc3Bxyc3MpQeJ+9uBZZ4XsZCMRclI1rK6upjOdyVQ0gfmNAInKYlkWMzMztB3u9XpRXl6e0RQXlmXR398Pr9cbV3lJLDLIQu10OjE5OUmz1OVy+XwF8oMPQtTKWQcPgn/GGdROiVRAyOgDec9FKR4LLSbvunXgFRam/VmTIYGxqpYejwdisRjT09Mpp/EAoS2Qj/uM1SlExc+JX2xFRQV+97vfpfQmWVlZGa9Mhr8HuY/J3KDf708pLzpRLJUIiHj9Wq1WTE9Pz1uhKRRomJ1F9uHD8J9xBuZKSuY/Y4yWNJcQRgpbWIp2NiHwk5OT1FVDoVCgqakJfD4fR44cgVAohM/nQ1ZWFgoKClBSUgKRSIS+vj4EAgFoNJqkFcapguvv63K5oNfr8cEHH0CpVKY0wpSuKjsnJwf19fUIBAIwmUzo7OykFoVFRUWLrkk6wp1Ez1MgENCCi91ux8DAAKxWK3g8Hq1WLhWWZWaSYZiQNnQ8uxxgvj1OIt3C5yMjIVKbOxgMoru7G4FAAOvXrw8hPvEqmQTT09MYGRmBQqFIyI8xfOHkJupIJBJ6Y3g8HnR0dECtVtNZw3AFNiFvRK2sVCpRWFgIycaN8Lz6atQK49TUFHp6ehZVDROtaEYC2f0Q78Hm5mZMT0+jt7cXHo8HcrkcSqUSeXl5KccLdnV1QSwWo6GhIekvuVQqhVarhVarDankjRcWYg0niYd/7rkIcn73IpEIZWVl0Gg0CAQCsNlsMBgMmJubo7YiRYODkC7M+RYIhch56CGgoSHpzxj+eVO5TuFVS3Kuyc5axjsfhmE+1j58p7AYLMv+J4D/SPd9wmcmM6HmZlmWfi9tNhtmZ2chlUohl8tpZOFSIl17IuDDTZbb7YbVaqUCoPz8fCgUCtqC5x08COlCN0QkEsHz3HOAXJ5QBTJSOxsMk7F2NhkVIiEfBQUFKC4uRm1tLQCEeP4KhUJ0d3ejoKCAWi6Rzh6JBtTr9RgcHKTdr6UkK1xIJBIqSDGbzdDr9Th69Ci0Wi2USmXCbedUhTTheg3il0nyyQcGBqBUKqHRaD70nU5RgJPqs0Imk6GsrAxSqZQ+uwszUAiJhmWPU0wEs7OzOHHiBEQiEWoSDKMPJ5OkLc217eEiEfX32NgYrUg6HI6EzoPsgLnzlevWraOLMsMwmJubw8mTJ7Fy5cqoaktu6Rr4sH3S3d0Nn883b5/zla/MR0JxXkcMw1tbWxd9sROpaEYDqRoKjxzBBoMBrN+PnI0bQ0iY0WhET08PsrOzqadlIq1jModSWFgIrVab8DlFA7eS5ygvx9FgEKV9fbA0NGBGIIBCr4dSqUR2dvYiw3RS0QRAK4CTu3YhZ8HzDQByjx0Dtm1L6dxI1UG8YsUiA/RUkZeXh9LS0qRmLcPBtRhaqjnMU/hkQCKRZKQySVqnNpsNdrsdExMTkMvlKCkpQW1t7bLeY+m0ucn6MDw8jOnpaWpDVlNTQ4kCF+EtadEHH4C95JL594rTko7UzsaZZy56bSQxTiSQosTU1BRsNhs999raWkrgA4EAXf/IdZqamoJYLMbMzAxcLhdYloVarYZMJqNrplgspoTOaDSira0NYrEYWq02o+k2scDj8VBcXIyhoSHU19fDYDCgv79/SdvOsUghyScPBoMwmUw4efIkAECtVqdUPY13vHggAs2ioqKMR5KGY1l9JhOZURwfH8fIyAiam5vR3t6e8PtzFz3iZVlXVxeVicdaJIn9kNPpxPr16zEzM4O5ubmEz4OQI6FQiNbWVkoueTweTCYT/XyRFqJoIEaqGo2G7vDHxsbQ3d2NnJwcFBYWwm63Y25ubpFhOEEwTkUzGkjsoqKvDzULaSn4+c9pm5zP59NYKO7sDfn9kUHqSKkTxKtTq9WiuDiSSDV1zM3N4cSJE2jYtg25ubmQA3QOVKfT0VECbrY5N+ZRIpFAq9VCcM01wJ//DNbnQ1AggL66GsaeHmqYnqgfGrfqUCEUYvyPfwQuvjitzxgMBunxI81aTk1NJVS1DM/lTubePIVPF8RiccozkySi02q1Ym5uDjKZDEKhEEVFRaiurl6qU46LZMkksRojsYVerxc5OTmorKwM+b4norD2bt6c8HEjtbPD5zWjiXHoeyw8HyYnJ2G322E0GqFUKmmKEXHBIM9jHo8XUm0NBoMoLCxEeXk5HfchcX9HjhxBQUEBysrK6GacYRiUlJSgtLQ0JN2mrKxsWUdmCJEjbWfyDCYimfBnz1JGFPJ4POpK4nQ6YTAYYDAYkJ+fT9foZI6XKplczsLAR9LmjoRgMIi+vj64XK6485GRQL4k4+PjGB0dRWtra8wHYrQ2N4lnzMvLQ0tLS0pG5AaDAVVVVSgrKwu58YaHhzE1NYU1a9akNDtBED5DNz09Tc1eZTIZxsfHoVQqI+7KiAIwUfh8PrS3t0OlUqF8eDhum5zrG1lZWQmv14vJyUkMDw/D4XBQBbZcLqet/qVQg5NWf1NTU4h9U6Q50MnJSYyOjoLP54cany/swL1r1yL44osQvPceZlpbIS4vp16R5HVEIR7LqoRbdQAA6aFDaZPJWKblAoEghOTHqlpy38duty+yvDqFfx9wo2/jrX0kEIGYhgPz1X21Wk2JiNlshtPpXJZzj4Z4ZJJlWczNzVEBCpnJJubn7e3ti2bXE1VYB5qbwfb1JXSe4a8FgPJnngE/GKRrbaTq5XRjI21fBwIB+jtwuVxYsWIF/fzcCiQJfZiamkJWVhaUSiUaGxsjjhxw4/4mJyfR399Pq5UKhYISM5lMhtWrV8Pn82FsbAwOhwMnT55cVsEOt+08NzdHLX2Ki4uhVqvp50unMpnM66RSKWpraykp5AqIVCpVXN1BOpZCnyo1N0GsNjfxKpTL5airqwu5cIn+4liWhdvthslkwsZgEMKHH44p8InU5nY4HGhvb6dxYtyfTYRMzs7OYnBwkO7cyE3Asiy6urogEAgyHunl9/sxNDSEsrIyaLXakHa41+ul1j55eXnxd1NhSm+Xy4WOjg5UVVVBqVSm1CYXiUT0i02G1ScnJ9HX1wePxwONRpNx4mKxWDA0NBSx1c9F+CgB2Z0PDQ3B6XTS2aC8vDzgtNPg2bABZp0OfJZFdnZ2CGHmGqaTGar8/PyQhSI8bcZz2mlIfUsxj0QXi2hVS6PRiN7e3vnzWvgOOZ3OpHbONpsN27Ztw8jICCoqKvDss88uGt9gGKYOwN84f1UF4Ecsy/6KYZgdAL4CwLLwb//DsuzehE/gFDIK7rxsJOGKx+Oh1UeHw4Hc3FwoFApoNJqIm+RMzF0C6alhI5HJ8Coq6fBE+hyRXp+owlr43nsoe/FF8K6+OqGNPHktIauVHg+YZ56hZDV8HelQKOAdGIBCocCqVasgFApp9ZFhGBiNRjqHT9ZfsrFXKpWoqalJWEjJMAzdoDqdTuj1egwNDUGlUtGMbdJaraiowNjYGIqKihIW7GS6kpmTk4NVq1ZRlfPx48eRlZUFjUaTMtFKh6CREQOXywWDwYB//etfKCgogEajiUq20zkeeS25rp/oBByCaBeDKJ0jxSuSama8KiWpnjEMg1a3G6JLL41rih5OJq1WK3p6eiJGKyayGJrNZgwMDKC6uhqzs7P0i0wqnVx7nkzB6XRSskf8v+K1w8ksY/g1DVd62559Fm0SCVatWvWhj2OKbXJ6DB4PcrkcLMvCZrOhubkZDocDXV1dH86BLpC3VG964k/Z2tqadERhVlYWja/iEl9iPcQwDAKBAIhiltw/XMN0AFTdOTg4SNWdCoUCEk7VYbi8HNnr16f0GblINU4xvGo5Pj4Ok8mE3/zmN/jLX/6C7OxsvP/++9i4cWPc79+9996L8847D9u3b8e9996Le++9F/fdd1/Iz7As2wugBQAYhuEDGAPwAudH/pdl2QeS/iCnsKQgHRwyczc1NbWoapdIyy9TivBU3SMIKSbVOJvNBpZlF1VRY70+nOgkorDmHTwI2ec+hzyvF/jTnxa1pGOBkFVeMDh/jH/8AxNaLazZ2RDcdx9KenrAP+88rDj3XHp+wWCQzuYxDIPKykqMjIxgcHAQQqEQJSUlqKysjDhylCykUinq6uoQCAQwMTGB9vZ2SKVSWv2bnJxEMBikc+iJCnaWgvBwVc4zMzPQ6/WYnJxEXl4eiouLk3pWpKsCB+af0ytWrEBNTQ0sFgv6+/vh8/lQVlaG4uLiUPeWNMkk2Rh9KhJwon0I0pKOlCQDJEYm7XY7Ojo6UF1djcHBQfD/+c+ETdG9Xi9YloV5zx4E9u3DxiiG4bFa9CzLYnh4GFarFevWrYPP58PExAQOHTqEnJwcTE9Px5zdTBXT09Po7u5GQ0ND1LSc8HY4yecmrVmiDpdKpYuU3tMvvojme+9dNCqQbJs8HEajETqdDmvWrIFIJIJCoQhRYIfPgSoUioRHAnQ6HSYnJxf5U6YCQnwJ+e3p6cHMzAxEIhGOHTtGF8jc3FywLBvSQpLJZJQQu91u2Gw29Pb2zvvOKRRQ3HgjZo1G5GSgQp2JNgbDMBAtZMF/97vfRUNDAx577DE89dRT+MY3voH3338/5iD7nj17sH//fgDADTfcgLPPPnsRmQzDeQAGWZYdTevET2HJIBAI8OCDD+KMM86gDg5yuRzl5eVJf7cyYRie6nuQCjzZ4JHoxdWrVydFICIdP2F/SJ9vPt0mrHrJRbTZS1JlZAUC9BYXQ+jxzJP4hoaQBDDueXLb11KpFKWlpWhoaMDU1BQMBgPcbjc0Gk1am3YuSIZ0Tk4ODAYD2tvbwbIsFAoFmpubowp22tvbIRKJQgQ7yzFjmZeXh7y8PAwNDWFubg5HjhyBTCaDRqNJyJcxk8Idrt0RMWc/ePAg8vPzoVarkZeXlzaZzKR9XywsW2WSi2AwiN7eXrjdbqxfvz7q4hRvzpJUAxsbG5GTk4PBwcGETdFJZXLkL3/Biq9/HTy/H/jznyNWMqPtrEm6DI/Hw5o1axAMBiESibBmzRqYTCb09fUhNzcX/f39MJlMET0jUwER8bS0tMR8yHPBMAz9EgEfWkQQax9NZSWqw5J4+EmIMBKJdyRm5JEEQuHEd25uDhaLBTqdDjwej4p4IlVCiHLe4XBkfIyAiLFYlsXGjRtpRdJms8FkMlFhCxHxiESiEBEPmdEkamvnvn0IPvYYfJWVGNywASqVCgqFImVrlFQrk/Hep7GxEffff39CC6fJZKJV2ZKSEpjNcWO1rwXwl7C/u4VhmC8COALgeyzLTiX5Ef4twTDM1QB2AKgHsIH7ML7nnnvw+OOPg8/n48EHH8RFF10U872mpqZw1113Yf/+/TAYDJidnYVarcbo6Cjq6upSPsdM2gvFA8uycDqdVEwSCARQUFAAmUyG4uLilGezo/pMxlFY+884AyKhEEGWBZNguo3xz3/GmFaLGYZB0a9/DfGBA1BefTWqTz89pPoIfCj4JF0UMp6jVCoXKeYlEglKSkpoZa6vry/E+DtZBINB2Gw2WCwWTE9PQyaT0eMGg0GMjY2hs7MTCoUCZWVlIdZ4XMGOXq+ngh2VSrVslmTkudLU1ITp6Wno9Xr09PTQYJFoz+l02uOxPhvXnJ2MW7nd7qiOL4ngUyfA4YKILhQKBVauXBnz4kYjk+G2O2SHyTDMvN1KAqbo5GZf09k5n4sao5IZaTEkKmRiQcC9UXQ6HcxmMzZu3Ehn0YjQg+sZqVQqk/LlYlkWo6OjsNlsWLt2bVqklJsmEwgEYK2qwrGf/xzZR44gcMYZkNbUQOHzJVQVjGeGzrIsBgYG4PF4EiJ7DMMgNzcXubm5qK6uhsfjoTGK3MWyoKAADMOgt7cXwWCQtp8zBTLrKhQKUV9fT987XL1OIiiJsIW064mNBqlaCg4fRsl//Afg86FEIIDp6acxpVCgq6uLDs0TQUyinyNTA9bham4yv0PO4/zzz4fRaFz0urvuuiup4zAMIwJwOYAfcP76twB2AmAX/vsLADcm+RH+XXECwOcAhDiUd3V14a9//StOnjyJ8fFxnH/++ejr64u58ZBKpTj33HPx05/+FKeffjruuOMOAPObwHSw1Ck64ek5EomEzg+STdrg4GBax49HZmOJceZefBHW3btRdM01kTfa+/cDC9ZjQY8HwbffRtFtt2HFihXAunU4tmIFClavBp8jnuGqr4F5x4zq6uq4YwfcOXFuJUypVEKtVsd9HhFBpcVigdPphFwuh0qlQl1d3aJ1qKqqChUVFbBYLOjp6aGG3qTbQ+a4SUyhwWDA0aNH4fF46AzrUoKcA8MwKCgoQEFBAbxeL8bGxnDo0CHk5ubSCm6k1yWLZObbSfHE4/Ggv78fZrMZJ06coNXK5X4+JIJlbXMHg0EcPnwYdXV1i+YjIyESmfT7/ejs7IREIlkUa0gWHF4cU/S5uTn09/cjNzcXeVu3Ao88ErOSGb6Qzc3NoaOjA7W1tTTSjiw2hNhwz437Ba6pqYHL5YLFYsHJkycRCARi5l8TBINB9PT0AEDGq28Mw8BisYB/+unQfOlLsNvttCpIlMpKpTKqOj6WGToxIxeJRDHNyGNVNsViccgsI0lt6Ovrg8/no0kEmSSSwWAQJ06cgEwmQ2VlZdT35gpbuBGUJMWItOsLCgogPnBgvuW1kMYjO3oUknPPpZsRm82G8fFx9PT00DSKeG3+TLUxAoEA3ZQ5HI5Fv+u33nor6mtVKhUmJiZQUlKCiYmJePm9nwVwjGVZE/kL7v8zDPMogFdS+hD/hmBZthtYPEq0Z88eXHvttTS/t6amBocOHcJpp50W9b3EYjG2bNnCfe+MrDORRDypvAeXzJHkKq75dqz0nEy0ULmvD69CxhLjBDZsgC4rC4Vr1tDX+vftg7m+HgaNBvmlpWgWiebXBpEIeVu3wpeTQ599hYWFaGtro0LB2dlZ2g1pbm5OejacgFTCKisrqZVOVlYWtFptCGFxOBywWCyYnJwEy7IJE1dg/tlJgjjsdjv0ej0GBgZoZVIgEFDCU1FRgbKyMhw6dAh9fX3w+/3QarVLlrATiRSKRCKad22z2TA0NASPx0MruNzzzcTx4kEsFkOlUkEoFKKwsBCjo6NwOp1U1Bqv2POpI5PAvAG42+3GunXr6JciHsLJpNPpRHt7O8rLy2lqTKSfj1WxM5lMGBwcRG1tLaxWK9g1a+LGO3LJJGmtNzU1UfsYhmEoySVzRbFuGuJfqNVqF+VfE4Ud8T0E5gVGib53siCemGR2MbwqGN4OJ7OC+fn59CaNpvLmvnd5eXnUc0gm5pEM/+fn56O9vR2FhYUQCoUhxJzMMqZ6nQKBQMg1SQbh1kOzs7OYnJyETqdDXlERmhdmoCASgXfOOQgsHA9ASI4vadV1dHQAACWW3MF53sGDKH32WQg//3kgCS+7aJ+Zaw2UTFXg8ssvx1NPPYXt27fjqaeewtatW2P9+HUIa3Ez85nSEwt/vBLz1bZTSANjY2PYxFnL1Go1xsbGEn69SCSC1+vNSKJJJtJnGIbB1NQU7HY7pqen6bw1ye5O5PXpnAN3LYlUhYwnxgkGg7BYLHC/8w7qbr4ZPJ8POSIRVC++CPYLX4Cjpga8f/4T/tNPR2DtWvAWninkM/N4PLhcLrjdbrqeZsoLluuJODMzQ59FWVlZ8Hg8kEgkMW2DEoVMJkN9fT38fj/Gx8dx7NgxmnOdk5ODYDAIt9sNoVCINWvWLHnCTixyxzAMXXO5FdyCggL4fL4lrUxGOk+ubR3x/Dx8+DBkMhnUajXt0sU65qdCgDMxMQGLxQK5XJ7ULopLJonaevXq1YvKzpF+Phzc1vj69evhdrvpbFe8eEeysx4eHobFYsHatWupypFhmBALnThVmUXgkg+iICbqLqlUiry8PExMTKCqqgoqlSqp944Ht9uNjo6OmIbh4e1wknTT29sLmUw2X45fswYIU3kTuye1Wk3n6aIh2ZhHn8+HtrY2OtsCgFYFbTZbSLY5EfEkOhLg9/vR3t6O4uJilJWVJfSaaODOqVZXV8PV2Iijfj/kHR2wNTeDJ5ejcHaWDn2TQXWWZal9BVecpNPpYLfbkZubi9LRUZTdcAMqPB4wf/5zUirRSAhvc0farEXD9u3bcc011+Dxxx+HVqvFc889Rz5/KYDHWJbdsvBnKYALANwU9hY/ZximBfNt7pEI//5vDYZh3gIQ6Qt6B8uyeyK9JhJxSuZhIhaL4Xa7M0YmU6lMkgg/IiYhaScktjAZZIJMktdHrEJ+73uLxDjEpo1Y8czOzqKyv39+rCoYBHw+8N97D5716xFYtw7M+vXzFdexMdhsNtru5FYBWZaFxWJBd3c3eDxextJm/H4/rFYrLBYLHA4H3Uy63W7aJclU5KVAIIBWq4VGo6GWak6nEwzDQCwWUycSIthZsWJFVMFOOki08s6dZbRYLJiYmKDRjcnMm2bKiojr+cmd9SwpKUFZWVkIx/rUCXCKi4tRWFiIzs7OhFJwCIg35cjICEwmE9atWxfzho5GJgOBADo7OyEWi2n72ev1JrzAEf89h8OB9X4/mF/+EoEzzwRz2mlUqRtLVZ0owhXEROAhFouh0+ngdDppDGC6XyS73U5TghId8A2fFSQimePHj4PH56Pw6qvnc1FdLrS3tydsRp6Mf6XH40FbWxsqKysXEXfxsWMoe/ddlJx1FgKbNtE51ZGREWqFQ9TrkUBIqkajyXgaj9/vR1dXF0q2bEHJl78MFcd6aHBwECKRiBLfrKwsSizJPapQKFBUVASGYTA7Owv+n/8M1uOhtiHBt98Gu2FDWtVYsug4HI6kvD8VCgX27du36O9Zlh0HsIXzZyeARTcEO58pfQpRwLLs+cm+Rq1WQ6/X0z8bDIakNgikKpUJJErkgsFgiAE6qcZUV1djZGQkphdfps4hHKSdLdNowC5sLqNVIf3r18NWV4fJyUlMHT4MRV8fSnp7oTrvPHRkZ8+Py5xzDvDrX4MFAJEIgQW1vNVqxfT0NI2i1Wq1EVuYXPXv3NwcdDodBgYGaBs2GeJAyK7FYqHWbBqNJqSrQ2IByfMzUcVzLBC/S7PZjKmpKWpnR/xL7XY78vLyIBaLKZEilVOuYKe0tBRlZWUpB4AkawZOrr1UKkVjYyPGx8fxwQcfUG/SeGtmphN3uLOePp8P4+PjOHr0KL2e3BG85cCykEkejweGiZ+CEw6GYaDT6SCTybB+/fq4rD7SDtjtdqOtrY1W1wiiJeCEgwhteDweVs3MQLzgYQmRCIYnn8SQSoU1a9ZkbNdGQCx8NmzYAIlEQgefBwcH4XK5qAiF225OFDabDX19fVi9enVS5tRcRBPJdHd3Y2ZmhhKfRHZjifpXEl9NMqvKRaRWef7GjXRONbxdH379CEklBu2ZBPFBLSsro1Va7sYB+HBhJ2buXOshIDQ/VyqVQnjppWAeewxBjwcQCjFcUQHL4cPU+oQ7JpEIwslkqvfFKXw8cPnll+P666/Hd7/7XYyPj6O/vx8bksiCz8rKovnc6SJWZTI8tpAYoIdbEKXbKk9lbpPbzq4TCGD929+Ac88NsQTynHYaTOXlsJ48SUlQYWEhaiwWyL773fn16JFHILn7bkxXVSF3wwZM794N/1tvYWLlSkwKhSiYmooqYomFnJwcNDQ0wOv1wmAw4NChQzGFNNyxm8nJSTqHV19fH3VUILwFrtfr0d/fT/0QE11jAoEArFYrzGYz5ubmkJeXF9EwnZDXEydOQCQSQa1WIz8/n5IirmCHCGXy8vKS4hXc65GqxY9UKsWKFStQXV0Ns9mMrq4uAIBGo0FRUVHE3+NSmqQLhUKUl5dDq9ViZmYGBoOB6je4qUdLiWVVc3NjuuLB7XZDr9cjNzc3pnCDi3CyOj09jZMnT2LVqlWLqm+REnDCQTwsa2pq0NfXN6+647Q32HfewZqf/zyjZWSWZalfIjd2kZskQ9SLpHJJLBkS8WQkPo+tra0ZJcBisRgSiQR+vx8bN26kaUTEOodUBaOdXzz/SpqzHaUCHK9VHt6u516/rKws2O32iMb56YJUO8kgeTRwzebJOIHZbEZvby+kUimNoCQJE7516zCzezcszz0H1bZtKN+0CeWYz6W3Wq0YHh6mc2Uk5jEWuBYSp8jkJwcMw1wJ4CEASgCvXnTRRXjjjTfQ0NCAa665BqtWrYJAIMD//d//JbVOcfO50wWXTJKOBjEOZxgGcrk8rgF6umQylddz29k8AKIPPgB7zjlwOp2YLC6G9Zxz5o3PXS5otVrIZDJq3cPj+h0DqDYY0N7fD7fbPW/R8+Uvo0SpRE0G0r9EIhFVThMhDZnLl8lk1L5nZmaGBlek4hdKRnY8Hg8lcoWFhVCr1RHXF6/XC4vFArPZDI/HA4VCEVeNHKkKGU5eCbki5ImMXh08eJB2lRIhbekmKpHzJWNqDocDBoMBAwMD1OWFe12WI3GHK/b1+/3417/+hZMnTyIrKwsNDQ0pi7USwbKquROtTBISWFpaSquaiYD7/mNjY9QcO9KNHm+Ox2KxoK+vjwptSkpK0FVUhBaBAAzLAgIBlFdfDTaBBToRD0bgw3zyQCCA1tbWqDcQUVgXFhYu8mTktqK5n5vYCpFs8HS9LsMRTlLJPCXLslQdfvz48biekQTca2atrUVvb++inG0ukmmVc6+f3W6nQh6dTgedTkfV9emmRJCqdmVlZVIkNZ71EKloDstkKPz+9xEsKUFw4b7Pzs5Gbm4uqqqq6MxZX18fvF5vSMxj+L3FFa6dIpOfHLAs+wJCk4QoY7rjjjuovU+yIDOTnOOk/F3w+/3weDzo6uqiDgeEWCTaokxXEZ6ItU9Ej8iFNSUoEEBXXQ3L4cPUeqihoYG2YgOBAC2UMAyDwBlngOX49prr61FVVYXs7GyMj4/DaDSCx+NBLBZnbC0mREwul0On06GtrY36bGq1WqxcuTIjyl6xWEzJK7GsIbOMQqGQts4BUN/JVCJzSSGJCE6OHDmCgoICqNXqEPGrXC6HVCpFU1MT9Ho9Dhw4gKKioqgklyCdezoSsrOzUVdXh5qaGkrqBQIBNBoNXctTbXOn0soXCAQQiURobW2Fw+FYUiIJfAwrkwaDAXq9HmvWrKHKvURByGRvby+cTmdKhuikMkhmNMlOqLy8HJ7rr8dxloWquxvWxkZM83goHBqKSTwSVSoTNXh+fj4qKioSvukiqa/JgLbX66WkaXx8HCzLorm5OeNWAdxKavj1ZhgGOTk5yMnJoQQnvF1PrHPIeXGvGSsUwnj//Wi5/vqYYoBUoh5JtbOpqYnOYpG228jICG1bEbP5ZCo7pG1eU1OTslEyENl6aHJyEidOzAuexWIxxGIxCgoKIBAIQgzTBQJBiGE6Nx6SPBDJYH06lcnnnnsOO3bsQHd3Nw4dOoR169ZF+ywXA/g1AD7mhTn3Lvy9HPO53RWYF+Bcc8q0/KNFVlYWvF4vgOSjDLkbIJvNRmd/E4ktjIZMCGiikdFI6mx3ayusWi3cv/oVpAcPYqqlBZLNm7FuIRKXVCB9Ph99fyKUm5ychEcohPqRR6Dq6YHw/PNRybFkqqurg8/nw9jYGA4fPkxdI1IVO5HrTex7gHkSt27dOjAMA4PBgL6+PhQXFydF4OOB2P5kZWXBYDCgra2NHnv16tUJB2rEA1dwQkaBWJalxabJyUlKnEhEodFoREdHB0QiETQaDRQKxaL7LtNkkoDP59Mu4tzcHK2uer1eeL3epDuCmcjmzs3NXXIhzrKSyViVSZKK4/F4sGHDBvD5fLhcroTb4sD8zUFsBFpaWuKat0Y6h+7ubgSDQaxdu5bG5BHRQ1dXF+o///n5Cg9AH+xDQ0PUwDV8jjERpXIiqupEQVTARBFnsVjQ0dFB/cEmJyehUCgycmMRM3K3252w92W4ZyRpwfT19VHvtLJ33vnwmrEsGiwWBBNYaJOJepyZmUF3d/eiaqdIJKJtlmAwSEU8RCRD2vWxFkoypxtptjNd8Hg8qu4vKysLqUpzzW7Jzp07M5Obm0ttuQip1z/7LHKPH4e4vh5zVVXIyclJmkyuXr0azz//PG66KboIm5nP4/4/zKu5DQAOMwzzEsuyXQC2A9jHsuy9DMNsX/jz7SldoFPICLKysuByuQAkRiaJjY3VasXs7CykUmlIbCExgU4V6RqfxyKj4eps09/+Br1QOC96u/xySK+9FrMDA8jv7obwb3+D/4wzEFi/HgzDUAsvm81GOx51dXXzIr8YDiFCoRAVFRXQarUwm83o7OwM8XiMB67zBxGxKJXKiL6TtbW1CAQCVKCRm5tLW+CpgDuGMzs7i9zcXBQVFaG+vh6BQIASy1gt8FRAWrh+vx8TExPo7e0FME9eV61aRdc7bquciJT6+voWCXYy5aMaCzk5OVi1ahX8fj/ee+89HD9+HGKxOClFeibI5HLEVC47mYw0h0NsZMJTcZIR7JB5hcLCQtTU1CR9buQciDKLlNB5PB6MRiPNEOd+MYRCYQjx4EbskdmUotNPhzBG+3V2dhYnT55EfX19iP9moq3xWAgGgzAYDKipqaED1BaLBUNDQxCLxbSVmsrsJDEjFwqFWL16dUo7PG7Lm7TDJycn0VNcjKaFcQJGJAL7mc8k/94xrh8RIDU3N8dc6Hg8HlXLrVixgopkuru74fP5qEgmLy+PftldC0r2lStXJuynmigCgQDa29uhUqmobRG3Kk1EWsPDw3A4HFQMQKq+4VVL7dgYZLfeCni9qBAI0CWT4fwHHgAAvPTSS7jssssSas/X19cncvobAAywLDsEAAzD/BXAVgBdC/89e+HnngKwH6fI5EcK7sxkJCIXLbawuLh4UYxfJpBJax8CMjvt0mhQJxCAx7JghUIUXnUVFC0t9Gf8fj/kvb0o+dKXwPP5wIpE6P2//8N4eXnIDGIqFT8yc6dSqTAzM4ORkRH4fD5otVoolcqQdZV4ElsslhBP4nARSyTw+XxoNBqo1Wo6+gIAWq02YtUuHNzkG5fLBblcjtLSUqxatSrktXw+P6QFfvLkSQiFQmg0mqheiPHg8Xjo7KXf74dCocCKFStowtjExAQ6OjoglUqhVquRm5tLn9/Z2dkhgp3Dhw9TMr1UlclIIJXTTQsuI0SRHsnOJxzpGo9nwrA/ESzrzKRAIIDD4Qj5N26aTPiDK1EySTwo1Wp1SvMnDocD7e3tqK6uRmFhYYicfnBwELOzs3HjC8OJ0ezsLCwWC0b4fMh/+UuU9vVBfNFFEHCIjcViweDgIJqbm0PsapIx8Y4Gonzm2vOQwdwVK1bA6XTCYrHgxIkTCAQCNOUmkTlBYupdUFCAioqKpM4rGkg7XCaTYSgYxMkHH0Rpby8m6upgZlkU9PRQM/e4qv4Y14+Q6VQESOEiGavViomJCZpYk5ubi/HxcaxatSqh6kIyiEQkw8EVaXGrqiTCk9yfWVlZ8zOSnIoMw7JYsWDMe9ZZZ8FoNOKqq67C17/+dVx77bWZ+AhlAPScPxsAkJtaRUzLWZadYBgmObPWU8g4uGpuQiYDgQCmp6epjU2k2MKlQqbIJIkhnJycpMk5hRdeCNfLL0Pw3nvwnX46/OvWAQvPHWIjJz18GDyfD8yCFVfusWMo/fznM9Yy5gonnE4n9Ho9hoaGUFhYCIFAQAk7UbqnMy7AnRcniTTEaYL7jCPPCIvFklLyDRGmhAtpSkpK4pJf0ra3WCxgGAZKpTKi6pzP50OtVqOsrIySNJfLRXO+aSregmCnvLyceltOT09DKpVCJpMtW0oM8KGQyefzUc/K7OzsqLZLy5likw6WrTIZyRrIaDRiaGgIzc3NEUvu8cgkmW80Go1Yt24dbDYbbc0kCkJEGxsbkZ2dTYkkNwYwXss80mclN0xNTQ1cTU30ixFYUMAFAgHMzMxg7dq1ixakZE28wzE9PY3u7m6sXr06qi+bVCqlXy7SricVLW72dfhNTCq4XMPwTIFlWRoZufILXwDDMKgGULkQoUja4VKplLabI+3ool0/o9FI53HTfQjw+Xzq98ayLMxmM3p6eiASidDf309FPJnwBA0EAmhra6MRZImAW1UFPrQe4lojla5Zg6yFqjkrEMC7eTP8fj+mp6dx++2343/+53/o+8XK5o6TeEMQ6SIs/Xb5FFICmZmcm5uD2+1Gd3c3AoEA8vPzafdnOR9wqba5ycbebDZjZmYGU1NTIck5pDXqXbsW3rVrAQDCI0cQfPttmFetgkGjgVAohHr9+nmB30Lkoe/003HkyBHaycpUMgvLsvD5fJRsmUwmBAIBKJVKVFZWZjQBhiTSkPnNQ4cOIScnBwKBADMzM7R7lW7yDVdIQ45DPC0JOeQWYCYnJ5GVlRW1bR8JXDJO1OZkHrWsrIz+rolgR6FQ4Pjx43A6nfjggw9QVFQUcj7LAaFQSM3bifl4d3c3fbaSZ1SmyOSn0hqIzNrNzs5i/fr1UR/s8WYsu7q6wLIs9aBM1sfS6/ViYGCAVh3Jzeb1etHR0YGSkpIQb8pUwY1P9Hq9OHHiBBwOBwQCAYYWBDyJxBMmArPZjOHh4UUt+VgIb9dHI26kOlZTU4PCwsKUrkU0kCxsqVSK6urq0PiyhQhFhUIRMmze3t4OALSqSohbpOs3NjYGo9GI1tbWpKrXiYwbOBwODA0NYe3atZDJZLQlNDQ0RMk5aTcnO6uaCpGMhPCq6tTUFMZYFkM/+xkUnZ2wNDQgt7QUv9q+HRs3bly0eMXK5k4QBgAazp/VAMYX/t/ELEQqMgxTAsCc7sH+ncAwzNUAdgCoP3z4MBVAjYyMoL6+HnV1dQCATZs24ZFHHon5Xl6vF++//z727dsHnU6Hnp4eXHjhhVSR+lEhmcokVwhDkrAkEgkkEgmqqqoWiWcIUZ2ZmVkUd1j68stgNm4E1qyBZ+9euhYUbdyIwgVPRNJiJRXDZEFmEMPteyoqKuhzKZW5ykRAPrfT6QQwP+/t9XohlUpRUVGRtkE5FyT3ury8nHbFWJaFSCSCy+Win7uysjKtmX6u2txisaCnpwc8Hg9qtRoKhQKBhVn8YDCIiooKSKVSKtghBC+R1n+mwDUfJ4SbtOPJyN2pymQY+Hw+fD4fjh8/DplMhjVr1sT8hUUjh8RypaioKCSrOlEyGQwG0dPTg0AgQM+BzE/Y7XacPHlyScQTgUAAXV1dyMvLQ2trK1iWjThnqVi7dlE8YSSEEx29Xg+z2ZxW5S0acTt27BgcDgfd5WUShKQWFhbGzcLmqpsrKysjE7fqahQutK2CZ52F4eJi2MxmtLS0JLVIJTJuwFWEEyFPeLuZDMoPDAwgKysrpN0c77pkgkiGgwgF7HY7PKedhvz/+A+88be/4bfbtsFms+HGG2/EwYMHQ7KdM4DDAFYwDFMJYAzAtQCuX/i3lwDcAODehf9GjAg8hag4AeBzAH4HIGTAuLq6mqpsE8GPf/xjzM3Noa6uDpdeeiluuukm9Pb2LrmtSDzEq0yS+c3JyUkEAgHI5XKqHmdZlirL/X4/nYX3er1w7tsH7N8Pa2MjhGedhYqBgZC4Q+H778O/kHsfLvAjQo/i4mJMTU1hYGAALMuivLw8rriCzAFaLBbqwVhaWor6+vpFr0tmrjIRkE4UiU6Uy+UoLi4OsQ6amZmBTqdDf38/NBoNbRmni0AgQI/t9/uRnZ1Nfyek6p0p1TFRm6tUKtjtduh0OvT29kIoFNIqu0AgAMuyCQl2lgOEcFdUVMBms2F4eBhWqxVisRjZ2dlp2Uh9aiqTpOJntVrR0NCA0tFR8O6/H8GzzgIb5aEVaQGJNWOZSCuEpJEUFBRAIpHQ5A8ej0dnGEnLO5PweDw0q5oQA+78Ssic5cgIhEIhlNdeC6VSiUiUI5zo9P/2t5heuTKmP2WyIMTN6/XCZDJhzZo1cDqd6F8w3yXqda4AJVlEytlOBuHEbWpqCpOTk+gHILnggvkfmppKyRIp3rgBUYSHz7yGvEdY0g3XM5IMkyuVypAIM2DpiCTB6OgoZmZm0NjYCAAYGBjAZz7zGdx77714++238fbbbydMJl944QV84xvfgMViwSWXXIKWlha88cYbYDjZ3CzL+hmGuQXAG5i3BnqCZdmTC29xL4BnGYb5fwB0AK7O9Of9NINl2W4gMw+Le+65BwDw+9//ntqypaukJkjXJJp7DtyZ4KmpKWRlZYXMb5L2NXEDkclkMBqNaGtrg1QqnV+/envR+J3vgPH5UL6wWcSFF87HHSbRFSKtU7lcDrvdjtHRUQwODoaQMK7f7uTkJHg8XtIejNHmKsn6F4touFyuD0etFmbkKysro87I5+XlobGxEW63GwaDAQcPHqRG3MluLIh5ucVigdvtpqMB3DWP2wKXy+XQaDRR19RkQMir2WyG3W6nFcepqSkqIsvJyUlIsJNuXHIyYBiGFnWOHTsGv9+PgwcPoqCgIK1Y0aXEspFJ4n0olUpROjoK4Wc/C3i94ItE8L32WkRCGX6Tm0wmKlhJZcaSCG1IZB7DMDhy5Aj9Qrnd7ogzjOlibm4ubrVz0Zzlwpf/5MmTEQUyIUTH40HO0aMou+qqjO8+TCYTRkdHqWCloKAAZWVltDVDBCi0qqpQJLx7crvdIb+PdMGtqhKbJ7vdDh6Ph6NHjyZtRh5r3GB6eho9PT1xFeHhyM7ORnZ2NsrLy+H3+2G1WmEwGDA7O0uvYX5+Pk6cOIHS0lIav5hJ6PV6TE1NoampCQzDYMeOHbDb7fj9738PPp+PK6+8EldeeWXC7xft5yNkc+8FsDfCz1kBnJfShzmFmBgeHkZraytyc3Pxs5/9DGeeeWZCr5NIJDCb56cNMkEmud2fVF/v9/thNBoxOTkJp9NJ3Qqqqqro+wcCgUXtayIacrvdkEqldNRKPTAAxucL2Sz6v//9pP1quSBxfx6PB6Ojo7QbQapwSqUSLS0taVd6pVLpIr9K7vwmN8yCRCcqlUo0NDQkNXeZlZWFmpoaVFZWwmg00q6iVquNSWjI88tsNoNlWSiVSqxYsSIqcQ5vgXd3d4PH4yVloUPg8/nosQl5DRctka7gyMgIvF4v1Gr1ogjgcMGOz+dLeewtXfFYeXk5Vq5cSX02/X4/3axEq+Qup1odWEYy6XQ6sW7duvkUlAgVn0CMKgjLshgaGsLU1FTKM5ZEaEPyqIPBILRaLdRqNTo7O6ly8cSJE2lZ5oSDmEQnW+3kzlmGC2TkcjnKWlqgFInms5lFIhRceSWCGb5xuG3zcIIYntKyqKoax4+RqM3r6uoWRV2mCyLkEQgE2LBhQ0hVnFxDYqsRa46RGKHzn34a4FxbYi3U0tKS1kC8QCCgbRiuUIDEX3m9XjgcjoxWyQ0GAyYnJ9Hc3AyGYXDPPffAaDTiySefXHJT21NIHQzDvAUgkgntHSzLRhwLKCkpoYlOR48exRVXXIGTJ08mVGHhqrnTTZ8BUosz5JqfG41GBINBlJaW0uhF4gPMPTc+n09HX6xWK63+V5lMyD12jBJEj8eDSYMB+QIBeEDIZjEZv9pwhLeQuZWwrKwsFBQUZHRkgOtXaTQacfToUQDz144ELpBkmnTA5/Np94i09IPBYMgsrd1uh9lshtVqhUAgQFFRUdLiHW5rmmv4HUltzoXH44HZbIbZbKaCpVjklVv5c7lctCqqVCpRVlZGq9tcwY7b7YZOp4PdbkdfX19Sgp1MeEUSVbtSqYTL5YLBYMCBAwfoJiL8sy73rOWykcny8nIEg8H5L/9ZZ4GfoMCEZVm0t7dDLBZjzZo1MS9ONDKp1+sxNjZGq47kJvH7/ejo6KCzesSE1mKxoLOzk+6oiLgjWRgMBkxMTGDNmjVpLSCR/CyHvV70/eQnKOvvh/iiiyBbuzbhX2Y8UQkxf3c6nQm1zSNVVbl+jOGtXDJnGEttniqCwSBOnjwJiUQSIuQJNyMPn2Mk5DfSwid45hnA64Xg6adhfuYZ9C/MvGbSDoW0WGZmZlBfX4+CgoL5dv3CSEGkpKBkMTY2BrPZTInkL37xCwwODuJPf/rTKSL5MQfLsucn+xqSjgQAa9euRXV1Nfr6+qImFIW/lmsNlK5PXaIpOsR+aHJyEjMzM9T8nIgXtVptSPuaYRg6626z2eh8WWFhIa3C8Q4ehHjbtpDZZ/HGjSi76iq4S0thf/VV6KqqIFYooF1IwUkGXAudYDAYsYVMRDTpinXC4ff7KXm12+00WnBqagoejwc8Hi+j0bnclr7D4cDAwAC6urrA4/GQl5eHoqKijJBX4EPDbxKnePjw4ZAWuNPphNlspvZBRUVFWLVqVdIz/RKJBDU1NaiqqoLJZKLxkGq1Gvn5+bTCRzLQp6amIJPJkhLspFMljEQKJRIJVqxYgerqapjNZnR1dQEANBoNioqKqK/wp5JMci8ku2kTfK+9RglNtJlJt9sNp9OJ8vLyhMrL4e0YUqHyeDx0ASVE0ul0orOzE9XV1SEtVq5lDpn16Ovrg8fjoa3m8Pm2cBC1usvlwpo1azL6oCZ5rna7HfUL9jkWiwW9R4/SiqBSqYxaMYsnKiHtYYFAgMbGxpS+AFzlMGnl6vV6zM3NISsrCw6HA83NzRknkoFAICSSMhoizTGSDQR5GJB2eHgV3fHqq2i9//6MCxL8fj+dHSWtbbVaDbVaHVFhT2ZtEyW0ExMTMBqNNKnooYceQnt7O/76179mPKf9FD4esFgsNAZ0aGgI/f39qKqqSui1WVlZ1LQ8WZeMSIjVKicZ8pOTk3TjVFRUhBUrVgCYX08tFgtmZ2fh9XqpgILMTHJjT6uqqhatt7Fmn5nNm5GzeTPqg0EYjUYcO3YMeXl5KC8vjzqzx7IsDYAg5JVECEZddzkimmTFOuEgkbkWiwU+ny9iG7e8vBxOpxM6nS7hucpEEJ5+k5eXh5UrV8LlcsFoNGJmZgZ5eXkZHRUjcYoajQZjY2M4duwYfD4fpFIpysrKErYPigduck64N2ZxcTHdEPD5/KQFO5moTEY7Z+LnSYJbBgYGqG0d93WfGgFOONhNm2K2tqenp2m7L1EBAnfR8/v9aG9vR15eHurq6ujf83g82qZsaGiISWhEIhGN/gsnRfn5+SgqKlpUKQoEAjhx4gSys7NTJmOxQNrmXNFHonOWQOyFlWtGzlXJpwNuK9dsNqO/vx9yuZz+bmNVBJMBUYQXFRUlPddC5hi52dckm7tMq0Ut8fsSCFB0zTXgLRGRVKvVEeM0IynsJycnKfklld9oRsZGoxHj4+OUSP7ud7/DBx98gF27di2rUvEUlgYMw1wJ4CEASq4A6t1338WPfvQjCAQC8Pl8PPLIIwk7VHCzuTPR5uZa+5B5PqKw5vP5UCgUqKmpoTGgLMtSlS8AKBQKzM3N4ciRI+Dz+dSEP5xEReq6JGK1xuPx6Izy5OQk9RiuqKhAbm4uDSkgpDY3NzcqeY13HbhiHZ1Oh4GBAWg0GhQXF0ckDeHiHTJiFMnEmwupVIqVK1dGnatMFJFa92VlZYvSb4gVD5l3LC8vTzn1hvvZyfiP1WqlHaesrCxMTEzAYDBQNXYmN8XEG9Nut2NwcBD9/f3g8/lQqVSoq6uj92aigp2lIpNcZGdno66uDitWrIDRaER3dzdcLhfMZjMNLllKMHHaFxkzFfb7/QgEAvjggw+wecFqIRrGxsag0+nQ0tKCzs5OtLa2JvTQY1kWBw4cQEtLC9ra2lBZWYmioiJajWQYhraem5qaUiYwpE1qNptpyZtULE+ePEl3KJnG2NgYxsfHE9qJRbJ/UCqVkPf2QnLZZYsqk4makaca8zg+Po6xsTG0tLTQ3yWpCE5OTtKEhVSMvokiXK1WZ1SwQn7PtldfhfCDD+A97TRIzj03rXna8OsXj0jGA4lYI556RJQgl8shEAhgMpmg1+upLdITTzyB1157Dc8//3xGDZCjYPmmv0+BICNr9qFDh/Cb3/wGDz30EMbHxxEMBtPy3O3o6IBcLsfc3BwVmykUCsjlcjp6xK1+ku+/3W6n2ddZWVmQy+Xw+/0wmUx0PIl8F2N1XVJZt8xmMwYHB+HxeCAUClFUVETdKzJZJPB4PNDr9bBYLDRej8/n01Ecm81GxTuFhYUpbwBJZU2n00EikaC8vDzq/Cy3+un3+5NKSANAK3Z2u52ubYmS7vDna05ODoqKiqBQKBa9ByHKExMT1A4q3Rlzt9tN2+fBYHA+FrmoCA6HA3q9fl68pVajsLDww00Mj0cJn81mg06ng9frpZsEj8eD7u5urFmzJunz+eCDD3Daaaclfc/Z7Xb09vZS0Vm8McEFpHxjLxuZJBmnBw4ciGiKDMyTwb6+PjidTjQ2NkIgEODo0aNJKdDeffdd8Pl8tLhckB05Av8ZZ9A2el9fH7xeL1atWpWx1jPZZROiJ5PJUFZWljEBDznG0NAQ7HY7Vq9enfS5kzlLi8WC6elpFA8Po6S3F+KLLgKzeTPNk45nRp5qzKNOp6Oij2jnTobmJycn46bwhL+ura0NFRUVKCrKfArf+Pg4JiYm0NzcPD+0v3COJNosVkUwHOHXz/nSSziyMHOjUqnSPldu+81ms9HqTl1dHZRKJf74xz9i9+7deOmll5Yr6eEUmVx+ZGTNbm9vxz333INHH30URqORzismA5fLRTc6s7OzUCqVKC0tRW5uLhXPBAIB+t0hFdCpqSlYrVaaP002R9y1I7hgGK7T6ZCbmztPjP7v/yDcuXO+68Lnw3fnnfB///uLzisasSQVQFIFIxVAqVQKk8kEh8MBrVa7qH2YKXg8HgwMDMBsNlOBSElJCeRyeUaPx7IspqenodPpQvwqyQwi17pIqVSmtVZ4vV4YDAaYTCYolUqo1eqIz/Lw9nm0zl+sz2SxWKDX68Hj8aDRaJIyHicVPHLtSZs40rkSWyabzQaVSoXS0lJa3CEepgzD0E2CyWRCQUEBdYtJFokU4CJhdnYWOp0Oq1evRjAYhFgsXlIyuextbpKCE15Z8/l86OjoQF5eXkh8YTLzOgaDAR6PB58RiSC7+mrA64Vw4aHdJpEgNzcXtbW1Gd1VEiHPzMwMNm7cCD6fT9MKSLWtqKgo5d0SmWHk8/nUyiVZhOeGzzU0wLBhA6xWK5hDh+B2u1FfXx831SbZmEeukIe0WKMhkl8kmRGMtit3u91oa2sLyR/PJAwGA8wcs3OBQEBtfUhFcHR0NGRmK/yhx0X49TM9+yy0P/hBRogkEOpDRzxT1Wo17rnnHvz973+H3+/H7373u1MzkqcQF9yZyUStgYLBIGZnZ6n3o0gkgkKhQF1dHUZGRlBaWors7Gz4fL6Qig6ZmbRarXRmuaKiImYVjGsYTnxbC8vKsEoonGfT0drZYRs618svY3LFCnrOZK0JF5EUFhZSNe/IyAjt4KRblAg3Ly8sLERrayvcbjf0ej0mJiYgEoky6nFIElfy8/NhNpsxNDSEzs5OSKVSaDSajM0gAqCilYqKCpoyQ5xKsrOzqQdkrPZ5op+JEEBu5nisWVGHw0ErkKSFnUjHktgyBQIBTExMoL29HVKpFGq1mm6UGIaBUChETU0NqqurMTIyApPJhGPHjkGj0YRUNZcKn1oBDkEkcsj1fwxv9QkEgrhkklQ0XS4XpFIpxP/8Z8hD2/i3v6HkBz9IqY0YD6QiyVX3cgU8XEUuqWQl2iYhanO5XJ6xGUaGYZCbm4vc3FwoFAp0d3ejtLQUOp0Ow8PDMdsZycQ8cnO2k50dDZ8RJPNCx48fpztmmUyG/v5+rFy5Evn5+Sldi1jQ6XSwWq1Rq6lCoZAOPhMDZULgyEB+eMoN9/oFBQJkXXQR8jNEJLkgNlLEveD0009Hd3c3brnlFuzZswePPvoodu/enfHjnsKnBxKJJCEy6fP5aHQh2VQpFAoaiUfa1wzDwGw2Q6vVgs/nY25uDlarlVrmpJoBTexSCgsLMVVdjY7//V/ktbUhe8sWZG/cuKgKGe7PO/7MM5i75RYUFRWhtrY25sM3KysLtbW18Pl8MBgMOHToEFQqFTQL+d2JgJsqNjk5Sc+/rq5ukeBHpVJhenoaQ0NDCAQCKC8vTzvmj2zUzWYzpqenkZOTg6qqKuTm5sJoNEKv18PlcmU0bxz4cCZVoVBAr9ejra2NdnfIXGqmyBU3c3x8fBxHjhxBfn4+HdMwmUyYnJyESCRCUVFRyr6ffD4farUaZWVlmJmZodeurKyMmtWTeUeFQgGHw4GKiooQwY5arV6yufXwWculJq/L1uYmWagdHR2orKykwhfi/9jY2Bhx99Xd3Y3i4uKoXoSEcOXk5KC6uhpHjhyBoq8PK2+5BYzPh6BAgKnnnoP0vMx6IpOqm8PhSKj1HD7ATWwUorUwSGKOVqtdEhJMzMibm5vpAh5tzpKbGx6tRcT9e//69VSERAyFMwW3242xsTGMjo5CLBYvyRzTyMgITYdJZWfndDrpdSQLZmFh4fyu9YMPYH7uuXki+dnPZuR8uSAGu62trRCJRHjppZfw8MMP45VXXlkS0h0Hp9rcy4+MrNlmsxnXXXcdXnzxRdhsNthsNtTU1NDUENK+JgKwwsJCyGSyiN6PpPo4NDQEm81GN4SkjRlv7Ux23nF2dhYjIyPIOn4cTd/9LhifDxAKMfL445ienkbTd78Lnt8PCIVwv/pqVDeReCCVKYPBQEWLkQgYaSuT8ROJREIJcKIkxuFwYHR0FHNzczHFOpFAxKMkBYao5bnrOgEZH9Dr9ZBKpRlJfiHzl2azmc4gKpVK8Hg86PV6WK1WOiuaaWJFBDyjo6NUvFRSUoLKysolIXEejwdjY2MwmUx0TMHlcmFiYgISiYRuWPx+P9URkAzuaHnrqba5LRYLpqamUFtbi2AwiKysrESekR//mUlCJk+ePImysjLk5eVBp9NRu5JoO9Le3l5KasLhcrnQ1taG8vJyqFQquoDNzc3B8tJLEL7/PrybN0N2wQVxB5eTWbBIxrZYLMaKFSuSJjFcn8NIw9V2ux0nTpxYEkNv4EMz8qampqjXJHzOUiaT0SHo8JZBeOuo45e/hPjss5OesUoEs7OzOHnyJBobGyGRSCIqLIn4JBVwZ1Mz0SIglRtyjj6fD6WlpUmrQBOBzWZDf38/JZKvvfYaHnjgAbz66qsZz5lPEKfI5PIjI2v23NwcLr74Yrz++uuw2WwwGAzIysrC9PQ0JBIJ7Rxwowu5zxKSKEba1wDo7OP09DTGx8cT8iRMdU47GAzCs2MHCv73f8ELBhHk8WD99rch+OEPIWlrSznhJhJYloXZbMbo6CikUikqKioWrU2JjMEkAq5Yp7i4OGpli9jamc1m2j4vKipKeL6bO1dJ0laSyQHnekByNw+RyLbf78fExATGxsaQl5cHjUYTMeEuUZDZcTL7mp2dDZVKhcLCQrhcLuj1eszMzFD1fqZJZTAYxOTkJHQ6HWZnZyEUCqFWq6mXNRBdsBNJrJQqmTSZTJibm0NNTc2nk0z29PRAoVDAZDKBZVk0NDTEfGgPDg4iOzt7UXVuamoKXV1dWLVqFXJzc6lim7zGbrfTWCsyVCwQCCL6MCazYHm9XnR0dND2RroIH/pmWRYejwcNDQ0ZnwPkzjAmQ5aiRXOR6yi4/346+B7k8TD1ne9A8tOfZvTcgfnfeW9vb8QIQ24Kj9VqTchzM/z1g4ODcLvdaGhoyHhLwOfz4fjx41AqlbRSQMyVEz3HWCDxjmTc4q233sLPfvYz7N27N+4sbCTceOONeOWVV1BUVIQTJ04s+neWZfGtb30Le/fuhVQqxZNPPkmViq+//jq+9a1voa+vbxDz2dz3pvXhTiEZZGTNNhgM+NznPoeysjJ87WtfA5/PR01NTcjmNpL6enZ2lravpVJp1ApcMBjE+Pg49Ho9tfiJVKXjri3xRDXYvx+2piYYNBrMzc2hdHQUdTffDMbnAysQ4Nj99yPv4otRUlKyZAKa0dFRqn4ngpP8/PyMryeBQIBWtgoKCujGnRC4dAM3uCB+ldPT01FnRbnPMW4LWalUJlx9ZVkWVqsVOp0OABIyA+e+dnp6GiaTKa4CHABtgU9MTCA/Pz9igkwyCAaDsFqtlMDJ5XKoVCrk5eVRFfjMzAyKi4tDZji5pJIr2FEqldScPVUyOTExAbfbjcrKyk8XmWRZFl6vF729vbBYLFCr1QnNAZJ4Pq7VDrEOIi1aQiRJ+klWVlbEiiE37J5bbs/7zW8SWrAcDgc6Ozvjqp5ThdFoxMjICIqKijA1NUVjoVKxywlHMBhET08PeDwe6urq0nov7nUMBALQGAyo/MpXKBn37t2bkR0/F5OTkzSXPRHiFX6OsZTXLMuiv78ffr8f9fX1S0IkSQWdqzgn7fDJycmISUGxwK2k2+rq0NPTQ+Md//GPf+DOO+/Eq6++mrK4591334VMJsMXv/jFiGRy7969eOihh7B3714cPHgQ3/rWt3Dw4EEEAgHU1tbi73//O6qrq8UADgO4jmXZrpRO5BQSAsMw9wO4rLGxsa66uhp/+MMf6FjDPffcg8cffxx8Ph8PPvggLrrooqjv09vbixtvvBFCoRAmkwmPPPIIampqoNfrsXLlykXta7/fT5NnnE5nwi4MBKStOjo6ijKdDprBQfDOPTfE0ifWRt/tdmPuzTeh/q//ou1s23PPQXLuufPKWs73xN3aSrsyZF4t3e4AmX+0WCwAQNfrYDCIkZERuN1ulJeXJ1XVSxRkkz8yMkJbuGVlZdBoNBlN5yLg2vAUFhZCrVbTYo3NZoNUKkVRUREKCwvTFvkRD87Z2dmoBDZ8/jPe6FgksCyLyclJ6PV6AEhKHENG18xmM+bm5qBQKKBSqaKu36S1PT4+jpycHKjVajruR6wLSdoUGTXg8/lwOBw488wzk75/xsbG4Pf7odVqwbLsp4tMWq1WHDlyBKWlpairq0vodcTXiVyQ/v5+OqdIjHBJ9nJHR0fCHo/cNkDW8eNovfVWMH5/1MokqYrFMzpPBSzLYnR0FDabDU1NTfSLSALrLRYLXC4X5HI5ioqKkp4PJMkweXl5qKioyOiiRhaYqb17UdTVheBZZyH7/PMjzuPEQqwxA/KwSXVQmiivSeQYN5ebx+Oht7cXANIm2dGOffz48bjWRaRaabFYMDc3F7Nlz33AskIhjv3856i8/npIJBK8//77uP322/HKK68kbPYfDSMjI7j00ksjksmbbroJZ599Nq677joA89du//79GBkZwY4dO/DGG28AAMMwzA8AgGXZe9I6mVOICYZhLgTwNsuyvttvvx0AcN9996GrqwvXXXcdDh06hPHxcZx//vno6+uLSqJI6lhBQQFaW1vxzjvvwO12o7OzE7W1tcjPz6eWP9z2dbobXuZf/wq5p+0vvgjhgsCPuzYENmwI6ZIIBALUvfAClA8+GLcYQOD3+6nfcCoCmvD0G0KgIhE4l8uF0dFRTE9PQ6PRpF0VJRU4QuCys7NpBc7hcGBkZCRjYp1IIONP5DOJxWJoNJqMEPNIIGIno9FI1d6khU4shFQqVdLPm0ggFcTp6WmUlJSgtLR00X1BCKTJZILdbo9LICOBZVk6OkJa20VFRRFb4LOzszh69CjEYjHlNok+A3U6HRiGgVqtBsuyiVo8ffytgTweD06cOAG1Wp3UronP58Pj8cDv96OzsxPZ2dlobm5GIBAAy7Lg8Xh0jq6uri7h2bCQdJvGRhhLShDYtw/G+noI8vOhtNmoe//ExAQ1fs600TPLsujt7UUgEFhknyMUCqm1AfHhGhsbQ3d3d8IzOKQqFs+MPFW43W5MTExg9X/+J7KzszE1NQWTyYTe3t6Yc5ZcxKo+kJ1cosb1kRCuvCbzqqQaSVIMlopIVlZWRpz55YKbFEQeWESVTVr2hYWFkEgkoYpUlkW92QxGIsHBgwdx22234aWXXlqS3zUXY2NjIWMearUaY2Nji/4egAFAZsvUp7AILMu+Sf5/06ZN2LVrFwBgz549uPbaayEWi1FZWYmamhocOnQIp512WsT3ycrKQlZWFgKBAPR6Pe6//37ccMMNMBqNAOY34bm5uSgtLU3ZQibSxpH/z3+C8fnABINg/X5Yn38ek3I5KisrkbVuHWxVVfOCnyeeQHF3N+TnngvtBRdAKBTOi2keeSQhlwlg/rtG4vkmJiZw9OhR6pgR6dlE1l6LxYKZmZmk0m8kEglWrlwJr9cLvV6PgwcPJi02iRRhSOImuc+L/Px8tLS0ULEOsQZLl8CGV+AKCgpQWVmJvLy8EHGLVqvNuOUNyb6WSCTQ6/XQ6/UQi8UoLy9HfX19RscVsrOzsXLlSlpBPHr0KPLy8lBaWgq32029RiMlLyUD4iGqUCjgcrkwNjaGQ4cOQalUoqysLKTbmpWVhezsbKxduxZjY2M4cuQIcnJyoNVqowp2CILBIL3HllrJDSwjmczKysLGjRthNBqp5UQi4PP5cLvdOHz4MFU2E6sJYjcxNDQUEi+YLAQCwbyy9rOfRcHCzstoNKK3t5ceh4gaMglSMczJyYlbFSMGukqlMmR3OjAwENJeCPdhJJZL8chMKuDOMJJrz7X0IRUEMqoQbYYxmn8lGTRvbW3N2K6X5HIXFBTgxIkT4PP5kEgk6OjoAJCZKguQHJEMB9cvksRkTk5Ooru7Gz6fD5qqKlQJhfNVeZEI/HPPxeGjR/Gd73wHL774YkZmeeMhUkeDG5kX/uNLfkKnQPHEE09g27ZtAOZJ/yaOWpmQ/nhwuVy4++678Zvf/Aa/+93v0NLSgl/96lcoLCzEyMgIjEYjJBJJ0mtitI1juO1Y4ec/j1mRCIcPH6bzfxUTE1Defvv8a5988sPXbtwIz6uvJi2qIdYupaWlMJvNaGtrQ05ODioqKiAQCKgjA+kKlZSUpDwGIxKJUF1djYqKCkoKwhN8uCDOGmazmR4/UQ/G7OxsrFq1KoTAxhLrRILf76edO5fLBYVCAY1Gs6gCV1BQgIKCAjpXOTg4mBEPTr/fTz+/0+mEQqFAbW0tcnNzqRn3xMQENBoNteHJFAQCAa0A6vV6HD16FEKhEBqNJmPCTAKJRIKamhpUVVXBZDLhxIkT1HM5EAjAZDLR9rRWq4VWq4XNZsPQ0BA8Hg9V9ke61ulEOKaCZfWZ5PF4EAgEcDqdCb+GyOpbWlqQl5cXIrQZGRmBzWajfnqZOkeiOjx58iSdNTh27FjIQHm6xyPxhalELxLT2YKCAupbZjabcfz4cfD5fBQVFUEqlaKvrw/19fVLYgljsVgwNDQUtVrL9bOsrq6OmRseyb9yeHgYs7Ozcc3OU0EwGMSJEycgk8lQVVUFAKisrKS+oESkFMkaKRGkQyQjQSKRQKPRQKPRzGfAFhfjwM6dKOrqwgtTU3C9+y6ee+45vPjii6ioqEj7eIlArVbTOSNgXrBRWlpKH2DcHwUwviwn9SkHwzBvAYjkE3YHy7J7AOCuu+6CQCDAF77wBQDRSX88eDwezM3N4W9/+xvKy8vxxBNP4Atf+AIuueQS3HzzzRCJRBgaGsLQ0BAqKysT7ghF2zgGNmzA1HPPwff3v2O8thazWVnQdnWhpqsLzvXr0ef1wvXaa1FDEwipTAU8Hg/FxcXIycnByMgIDh48CIZhUFxcjOrq6rQ3llzw+XxotVqo1WqYTCZKYMvLy8Hn8ymBI2tkOsfnElhSaSNinUgtT+7ol9frTer43Bxw4sGZbA44Gesym81wu90oLCxEZWXlIs/jvLw8NDY2wu12w2Aw4ODBg1CpVFCr1WkVfAKBACwWC0wmE1wuFwoLC7Fy5UrIZDKaepNKZTkR8Hg8qFQq8Pl8GAwGnDhxgnpzlpeXU8cEUgyRy+VUsHPgwIEQwQ7BcpPJZZuZBD6MzLNYLFi5cmXcnx8fH8fg4CDy8vLQ0NBAHd1ZlqVh8nV1dRm/YMQPkyQhAItV1wKBgKrV4rXtw9s6TqcTHR0dS5Lc4na7MTo6irGxMUgkEqhUKprAk6kFMVLOdjII97MsKCiAWq9H3vHjCJ51FvoUCng8HqxatSpjc5cEwWAwZH40GgKBAE3hIdZISqUSCoUi5mcm8Y6ZIpLhcDgc6OjoQGNjI7Kzs/HMM8/gwQcfBMMwKCkpwVe+8hVcddVVGTlWrJnJV199FQ8//DAV4Hzzm9/EoUOH4Pf7UVtbi3379qGqqooIcK5nWfZkRk7qFKKCYZgbNm3a9OS+ffvoQ+Wee+ZHVX/wgx8AAC666CLs2LEjaps7FjweD5544gn89re/xWc/+1nccsstEIvFGB4ehsfjQVVVFR0NIlhkGh5WmbT85S8Y02oxNTUV4r+Ydfz4ogqmw+FA/lVXzQttMiD0i+QAQTo8LpcLIyMjAOY3mkvl0Wq32zEyMgKLxUK/w9HIXrogkYPEo5e09UmMIKkAk2JEOkjUr5IQWJPJBJ/PB6VSCZVKlZSyOhAIwGg0wmAwQCaTQavVJqxrIBVQLoFUqVRR05e4Iprc3Fxotdq0bIzCVeDcGUyu4rygoABqtZr+XmIJdsi4QX9/PwoKCqhYOEFS//EX4ADzN87U1BTGxsbQ0NAQ/aAsi4GBAczNzaGiogKDg4NYvXo1hEIhJXpFRUXQaDQZnwUgRK+6ujomGSBZnlwbhkhfwvDF0/q3v6FdKsXq1aszLuQB5q0hhoeHaXJLeKuAVNpSvW5coVAmWs/hijxiYRDLAzMSErF3CgQC6OjogEKhSMoDk9uy52b2hufWLjWRdDqdaG9vR2NjI2QyGXp6evBf//Vf+Mtf/oKGhgbodDqYzWasW7cu7WNdd9112L9/PyYnJ6FSqfCTn/wEPp8PAPC1r30NLMvilltuweuvvw6pVIo//OEP9Lh79+7Ft7/9bfT39w8BeIJl2bvSPqFTiAmGYS4G8Euz2VzPvfdOnjyJ66+/ngpwzjvvPPT396f13fV6vfjDH/6A3/zmN7jwwgvxjW98AxKJBMPDw9SKRC6Xg3/o0KLvpGfNGtj//ncE334bxvp68DZvpmI47jlFswTiHTwI/1tvYbSyEtbaWtTZbChob0+4vR3un5uTk0M3iZHmuokButfrRUVFRdqiFkJgSVEiKytrEYH1+XwoLy9fssg9h8NBPZ4BoLi4eH42NcN6AOBDwdDo6CgCgQAleuEm5pkgsCzLYmpqCqOjowgGg1G9McNb+FwCm4yIxmq1Qq/XxzxWJJB70GQyYXZ2FnK5HMXFxVFFPFzFOcuyUKvVIfcGV7BDVPDT09Pg8/khI26fOjI5NzdHLV4igcwRSiQSrFixAn6/n1of8Hg8uN1urFixYklSYaanp9Hd3Y2GhoakXP+9Xi8llqQ9UFRUBJlMBuEDD4QsikNf+hLy7713SXaeBoMBJpMpIhGLNERObBQSebAQ5WtULAAAYGdJREFUH0aXyxXXGzQVEFsnMnScrFdkPD+6QCCA9vZ2FBUV0VitVEESHSwWC7X0yc/Px+DgIKqrq5fENooY9JNNSH9/P/7zP/8Tf/rTn6J+lz4GOGVavkxgGGYAgLi5uVkNzItwHnnkEQDzre8nnngCAoEAv/rVr/DZDCUveb1ePPXUU3j44YdxwQUX4Bvf+AakUimGh4fhcrnQvHcv8n7xC/qdHPl//w8j115LR1xiCRgS2Rx69u9H7uc+B57PB3bhZyKl2SSS7BUPRNRit9uh1WqhUqkSJh6RIgxjeSCmk3YTCWQzTDwgCYFVKpXwer3Q6XSYmZmJOX+XLlwuF1XQ+/1+KBQK1NTUpO2BGQ1cb8zS0lJqt2cymaiJO6lAZuJYer0eNpstaguc3ANGo5ESSOJDmcyGgXsslUoVcixSqWQYBoFAAEePHoXX60VeXt4ij9gY+GSQSZ/PB4fDge7ubmpwzIXb7UZbWxtVoJH5SIZhYLVa0dvbC4VCgdnZ2biu+snCaDRCp9OhqakprffjDg47HA6o9Xqs+PrXabSj6+WXwTv99LTPlwuWZUOSW+ItBtyEAG68l1KpjFgNJDnbDMMsiX0OaT3n5uaisrKS/n24V2Ss3PDwh4/35z8HY7UieNZZ8K5di/b2dmr3kEn4/X6YzWZqt0IeVNEeFKnA5XKhvb2dGvSPjIzguuuuwxNPPIG1a9cm/X7EVDwQCODLX/4ytm/fHvLv999/P55++mn6+bq7u2GxWCCXy1FRUYGcnBzw+XwIBAIcOXIk1qFOkcnlx7KLnXw+H17/8Y8x/Ic/gP3MZ3DpXXehs7MTqqEhbPjBD6jlmv3FFyE488yE3zfe2Er4BnLoS18C84MfQKVShWz4iM8s2eCns36RMaKpqSkq3olE9JKJMIwGj8cDnU6HyclJOlufqH9juIVQPGcNMutsNpuTtkuKBmLhYzab6Sx/UVEReDwetfshxu5LURX1+XwheeMkh3wpNvtAaJJPbm4u1Go1fD4fTCYTZmZmUFBQQG2M0n2GkijPsbExSKVSqNXqEHU3j8dDV1cXtFotdSyJp/5ewCeHTBKF8fr160P+bWZmBidOnEB9fX2I0IZhGOj1ehiNxhArikh5n6mUylmWxcjICKanp9HY2Ji22SoXZB7C/NJLyDlyBMGzzoLsggsSMlVNNN6RzI8yDIOVK1cmfZMSAQ9ZeMNbuESsshQ528CHFUOibIyGSHOW4YszuWasQgHRbbdRYtn2wAPIvegilJSUZPTcgflF+Pjx46iuroZCoVjkQUdnwFJcLMkGi3wv9Ho9tm3bht/97nfYmMK8GNdUXK1WY/369fjLX/6CVatWRfz5l19+Gf/7v/+Lt99+GwBQUVFBlagJ4BSZXH4sv3L+wAGw550H1uOBF8BlEgnmVq/G3XffjRaXC869e2FetQqKSy9NuXUbaT0M30Cann4a3fn5sNvtkEgkKCsrg0qlWhKiwiVfhOgFg8EQAQshsKlayBBwfTHJHH8koQmpfhHykoqJNzdvPD8/H1qtNqlnKtEVWCwWCIVCqFSqqCk44XOVxG4nHRARj8lkgtfrpS1sqVQKi8UCnU4HgUAArVa7aL43EyAekjqdDlNTU1QZXlFRsSQVX1IYIoS5rKwMubm5sFgsGB8fx7p16+bt5Hi8RC0ZPzlk0u/34+DBgyED4Eajkdr7ZGVlUSLJsiz6+vpoMkm0X0YkFVoiMxDBYHBJhTzcRJ7q6uqQaiAxm42UFpBovCMZCSAVvUx8McJbuD6fj3qaZfqL5/f70dbWRr00EwVZNEmQPRHHkGsZHu84/d3vIusnP8nouQMfEsmampqIQiqn00nNleNVViPB4/Hg+PHjWLlyJfLz8zE+Po6rr74aDz30EM4444yUzvnAgQNcU/FFAo1wXH/99TjnnHPwla98BcApMvkJwPKTyXvuAXvnnfT7FtixA09rNPj1r3+Ns846C9/61reQl5cH8549EB84AMlnP4vciy5KeD2JaicUDMLx1lvwv/UWxuvqEFi/HkVFRcjNzcX4+DgsFgs0Gk3U6mEmQMa2rFYrtXQpKSlJe/4vEoLBIO2g5eXlUQFNuAdkshXQSCBiHZ1OB5FIhPLy8oiVLSJMNZlMtIVOcrCTMYKfmpqCTqejc5XJbDp8Ph+tgHq9XloBjdZCn5ubg06ng91uj5iFnSzCoxzz8vJQXFyM/Px8uN1u2pYuLi5OynQ8GbjdboyNjWF8fJwWXqqrqzE+Po5du3ahoaEBO3bsSOStPhlk0u/3IxAI0KxJMoc3MzODpqamkEQbYlJeUFCQVGpLJH8sssBw38Pn86Gzs5OKMZbCsJorFOIiXo5pInm0Pp+Ptm6TtRZK9PyPHTtGVWWpzhpFAxGrlJeXpxz5B0TODdcYDNB+6UvUamgp4h3jEclwRKqsxoqdI0Syrq4OBQUFMBqNuPrqq/GLX/wCZ599dsrnvWvXLrz++ut47LHHAAB/+tOfcPDgQTz88MOLftbpdEKtVmNgYIBav1RWVtId/U033YSvfvWrsQ53ikwuPz6SyiTOO4+SPezbB5x2Gvx+P5555hn86le/wheqqvCdV1+lOdnHH3gA8ksuCUn+iIbw9XDqO99B/1VX0fZxtO+Rz+ejWcckoCLd6lCktZuMtVitVhgMBhQUFKC8vHxJKqLA/NozPDyMiYkJsCxL58CTSWFJBkRAQ4RBCoWCzmBardaMxihyU2hi+VVyVeB+vz+kApkoPB4PDAYDzGZz0u12UhE0mUyw2WzIy8uDSqWKup6TFvj4+HjSivNo8Hq9MJlMMJlMAACVSgW5XI4PPvgAf/7zn3HgwAGIRCJ84xvfwC233JIouf/4J+CEIxAIUIPO1tZWBINBSiRdLhc6OztRUVGRNNEQCAQoKSlBSUkJde7X6/Uhu7asrCx0dnaisrIyZsRdqnC5XOjo6Ij6/gzDICcnBzk5OaiurqazJe3t7fPmpLW1KBcK558KERIdltqMPNL7EwUaMXMnKRCpzAZ6PB60tbVlRKwSyc9yIj8f7+/cieLubjDnnAPZqlWQLdxbmUCyRBKYT3Ig9yU3haevry/Ev1QkElGiXVtbi4KCAlgsFlx99dW499570yKSQHK+gy+//DJOP/30EA/B999/n5o8X3DBBVi5ciXOipM4cgqfcpx22jyB3L8fOPvs+T8fOADB/v344tln4/pDh9B5/fUIut30gVNvMqFnagojIyM0ajQ8S5tsAN2bNkGwYNLPCgSwNTUllEAiFApRVVUFrVZLvQ+Li4uh0WiSIj3c7yu3q1ReXh7yPmq1GmVlZTCZTOjo6IBMJkNFRUVGqpQej4cWSfx+PwoLC7Fu3ToqUB0cHER5eTnkcnnGCWV+fj7y8vJgMpkwMDBAP1t5eXlCKUDJgKTQcP0qCdHj8Xi0Aun3+1FUVISVK1emfH3FYjGqq6tRWVkJo9GIjo4OSCSSqOkyRIlvNBphs9mQm5sLlUq1KIkoEgQCAY2dtNlsGBwchN/vpyrwRIszpAprMpkQCASgUqmwatUqHD16FA888AA++OADnHnmmbj55pvx17/+FT09PXjsscfg8/ky6osZCR9JZfK9996b/6KbTCjq6oL/jDPAbtoEhmEwNTWFnp4erFq1KtGB0YRAWqN6vR5WqxUKhQJqtTqpeZJEMDc3hxMnTqR8/mTRcL39NnKOHgX7mc8g58ILacvebrejs7NzyczIHQ4HOjs7aUUsEsLzabn2FvFK+ERMEuv90wGZMaytrUVOTk5S1cBwRHqwESKcDJGMBe7M6uTkJFiWhcfjQWVlJdRqNaxWKz7/+c9jx44d2LJlS9rHS6bNfeWVV+Lqq6/G9ddfH/G9duzYAZlMhltvvTXa4U5VJpcfH33aUKRKJfDhXCXL4sGtW/G5+++HXC6nOc+1VitKv/jF+dcJhRh69FHoFzKfNQYDlF1d4J97bspdhkAgQCM/i4qKoNVqoz5gSSHCYrGERBgm+rwg1jEjIyMQiUSorKxMuhLFtZ8DQHUBkZxA7HY7RkdH4XA4oNVqqdAlHRASbTKZMD09TcmTTCbD+Pg4TCZTRszCY8HtdmNwcBBms3lZfDinp6eh0+ng9Xppu5208a1WK3JycqBSqaBQKNK+vi6Xi/KRWC1w0m01Go0hbfyuri7s2rUL77zzDtatW4frrrsO5557brqk8ZPT5p6amsK//vUvbGJZFF57bcgcjEGjgcFgSFtRHQ3Eg7GxsREej2eR0q2wsDCtXZbVakV/fz+ampoyshsNj9TKzs7GzMwMmpubl8SjkmScJ+uBSRJ4JicnwTBM1EWPEFUiJsk0CFElM4ZcxJuzDEekOS1XSwva2tqwYsWKhBM/koHP56MpFcePH8c999wDPp+PL37xi7jtttsyIg7jmoqXlZVh/fr1eOaZZxb5vs7MzKCyshJ6vZ7OHjkcDgSDQeTk5MDhcOCCCy7Aj370I1x88cXRDneKTC4/Pnoyec89wJ13AoEAwOcDO3cCP/jBPMncvx+BM8/EcwYD3tixA5fn5mLdrbdCfPbZmL79dtT88Y/gBYNgeTzYvvMdCO68M+MVlWAwiImJCej1ejrmJBaLF623xJc3WfuWcEwtVGCB+ZnjWJtobgs9mWAMArfbDZ1OB5vNllKsYbgHYn5+PoqKiiJuwNMV60QDad8ScS25BuSzpTJXmQyIryNxSMnOzqbjWEsxe8tVZstkMmg0GmRnZ1MzdafTSdv4IyMjeO655/DGG2+gvr4e119/PS6++OKE748E8MkgkxMTE+jp6UEwGMQZ//wnRD/7GZ2DMf73f2PkuusSsrZJFizLUrPtxsbGkMUp3INLIpFEzLmOB5IKw1WcZxJGoxEDAwPIzc2dT4KI8SVPBSRnO10i7Ha76YLs8/mo5ybLsjh58iQ13M40iNl8IkQ13IRcIBAs8rMMn9Ny/8//4F/nnLOkRLKtrQ0VFRVQKpWYnZ3FVVddhTPPPBM2mw0HDhzAk08+GdFSK1kQU/FAIIAbb7wRd9xxB/Uk/NrXvgYAePLJJ/H666/jr3/9K33d0NAQrrzySgDzpPT666/HHXfcEetQp8jk8mN5yeQCQaTtbfJ3EWYow1/Hnncegm43PCyL65RKXHTRRfjarl3URm3iT39C/mc/u2TimWAwCIPBQIkemV3PdGIYwdzcHIaHh+H1eqkpOYAQE/NUnz/hIG1io9EY1+qHVGGJiEcul1MRTyLXgJCv0dFRCIVClJeXJ901I8UdLoGMVoUlpuszMzMZyQEnn4ErJMrOzqY+kGazGWNjY8jLy4NGo1mS5xcw/3vQ6XTQ6/Xw+/1gGAaNjY2w2Wx4/vnn8corr0Cr1WLbtm247LLLluo8Phlkcm5uDgDmhSMjI6j6yleAhYVj6Pe/R+nnP78kHoa9vb0IBoOor6+PuzBF2hkWFRVFJYjE43Fubg6NjY1LIv8ni0JzczOEQiFtP5jNZlplS6ayGt6+5eZsZ3CHQz03x8bGMD09DZVKhdLS0owIeLggEYPJms0ThPtZKhQKlOl0KLj6avpAPHrffSi+8solIZJ+vx/Hjx+nZsh2ux1XX301vva1r+G6664DADpTvBT31xLiFJlcfiwfmYxFGiORTC7uuQfBO+4Aj2XB8njovOYafHlwEFeoVPjyihWQXHwxhlQqWK1WaLXatI27CaJZoTEMg4mJiYzOOUaDw+FAX18f9UvOz8+nrdNMf7+51UOuMIj4YJpMJjgcjpAYv3SewTMzMxgdHaVt4lipMIRAmkwmsCxLo3+TyfLm+lVqNJqknl/kXjAajZicnIRUKqVK9PDfAxlb0Ol0AACtVpt2GhJ5X1IJnpmZoWbmVqsVP/nJT/Duu+8iLy8Pt9xyC2644YYli/Xk4JNBJicnJyEWiynJcLz1FsT/+hd8p5+O4iuuyDjTJorw/Pz8pBThBGRmhcxrEGJJbnZiLcTn85fEzJtlWQwPD2Nubi5qxZZbWSXehrHmF8Pbt2NPPYUhlYoS1UyD2/on1zORFIhEQWZIM1Xx5Kqu+YcPo6SnB2MrVqD085/PeI46ML/YHz9+nFpUOJ1OXHPNNbjhhhtwww03ZPx4y4xTZHL5EXPN/v73v4+XX34ZIpEI1dXV+MMf/kAfUPfccw8ef/xx8Pl8PPjgg7joootiHylaOzscCVYvgxs34oUXXsDPf/5zrF69GrfeeitKSkqocbdGo0FJSUnSpDJ8xjtaSAOpsI2MjEAikaCioiJjzyTSPjabzdQDsqCgALOzs3EN0DMBlmUxMTGBoaEhBINBWiiJlUOdDrgJNCSEhM/nw+1202cqgEXP1FSQrF8lqUBaLBZIJBLqhZnoc4hEFs7OzqZUGSX3o9FoxNTUFN1MeDwevPjii3jhhRcgEAhwzTXXYMuWLfj73/+OZ599Fq+99lpGfbCj4JNBJu+880689NJLuPDCC1FTU4M9e/bgkUceoQqlRILWEwVRJJeXl2ckepFbhifVq6mpKRQWFqK8vHxJiGRPTw9YlkV9fX3C78+dX4yUEhTevh398pehuP/+Jal4WSwWDA8Po6WlJYTYhufTEgIczdw2GojYqampaUliuVwuF44ePQqZTAaXyxV3zjJZBAIB6rNZUlICl8uFa6+9Ftdccw31dUwW8dJt9u/fj61bt9Kkoc997nP40Y9+lNBrU8ApMrn8iLlmv/nmmzj33HMhEAhw++23AwDuu+8+dHV14brrrqMZ3ueffz5NdQKQVjs72eplMBjEnj17cN9996G+vh7f//73UVZWhtHR0YS9I8MzuJNxnyC+h8PDwxAKhSmJZ4DFKTjR2sc+nw86nQ5msxklJSVQq9UZIw2RPJizsrJgMpnAMAwqKioyksgSDT6fDyMjIxgfHwePx0NWVhaKi4sTishNFuHZ3Ny5SofDQQmkWCymBDKd6+z1ejE2Ngaj0YjCwkKo1eqowiDyzCNCntzcXMpLXn75ZezatQtutxtXXXUVrrnmGpSVlS3Z7yQOPhlkEphn9T/+8Y/xpz/9CRUVFdi8eTO2bt2K9evX050hGTolJd9kS+9ESLJUime73Y62tjYIBAIwDBOSxZ2JG4DYJslksrRSZ7gpQYFAAEqlEiUjI7R9GxQI4Hn11chtqDRBWistLS1xK57cthMR8CiVypitppmZGXR3d2dM7BQOogonqvNE5iyTAUn+KS4uRmlpKTweD77whS/g0ksvxde//vWUfueJpNvs378fDzzwAF555ZWkX5sCTpHJ5UfCa/YLL7yAXbt24emnn16k6r/ooouwY8eO+XCJNNvZCVUvIyAYDOLll1/Gvffei9raWtx6663QarWUeBErHkIqScfLYrFQ8pauL+709DSGh4cBAFVVVXHnsUkCi9lshtvtps+GRFJw/H4/NZ4m/sSpzN+HFz6ieTDOzc1hZGQEbrcb5eXlMVvSycLtdlMRDY/Ho/OhExMTyMvLg1arXbJcbmD+mTI0NASbzQYejweZTEZJbKare8FgEGazGXq9HmKxmFoLEfcV0kbPzs5GcXExRCIR9u7di927d8NiseCKK67AtddeuyQJcyngk+MzefjwYXR3d6O/vx8ikQhvvvkmnnjiCXzzm9/EmWeeiSuuuIKm43A9IhMdCrZYLBgcHERzc/OSkAy73U6tf+RyOV3AhoeH6exJUVFRygpAYkZeXFwMtVqd1rlmZWVBo9FAo9HQRa6noADYuROq7m7kXn45pJs2ZfyJbzAYYDKZ0NramtAXNzs7G9nZ2aioqKDWSL29vVEjyaanp9HT04Pm5uYlsYgIJ5JAZD9Li8WCkydPJpz9S2ZV/WecgY4FO6XS0lJ4vV7ccMMNuPDCC1MmkgBw6NAh1NTUoKqqCgBw7bXXYs+ePQkRwnReewqfTDzxxBPYtm0bAGBsbAybNm2a/4cDB/BVqxWut9+eJ4j7988TyUBg/r/7939IHE87LfZm9Oyz5wkoIaJJ+KTyeDxs3boVl112GV599VXcdNNNqK6uxve//32sW7cOer0eBw4cQE5ODvx+PxX8JeJBmSjy8/PR2tqK2dlZDA8PIxAIUON+gkjkbcWKFUmTJYFAgPLycmg0GkxMTODYsWMJG6Bz28fEyHzVqlUx18ecnBw0NjbC5XJhdHQUQ0NDKY8SAKFjYTweD0VFRWhqagqZY9RqtZicnERPTw/9vJks+DidTlqBJFVlUp0lwRuZJpM8Hg/FxcUoLi7GzMwM1VAwDAOZTIbS0lIUFxfjrbfews9//nMMDQ3h0ksvxf33349Vq1Z9HAhkRrDslclgMEhnNrjweDzYt28fnnvuORw+fBibN2/GFVdcgdNPPx18Pj9k3oTMGITvOHU6HSwWC5qampZk/o8onlevXh1xliYQCNDznJ2dTTrairTml8pMneRsSyQSmt9JzNwT9V+Mlxk+OjqKqampjIiRwttEBQUFyMrKwsTEBFpbW5fEPioSkYwHn89HzzOanyV3VjUoEMDw5JNQXn45fD4fbrzxRmzcuBHf//7301pYEkm32b9/Pz7/+c/TGa0HHngADQ0NSSXjJIFPxyr5yQJ7/vnnw2g0LvqHu+66C1u3bqX/f+TIETz//PNgGAY333wzTjvtNPxHdTVw3nkIuN1ghUII9u+ff3G8dnYsxKteJohgMIjXXnsNd955JyQSCWZnZ3HrrbfSzZ1Go8lIyk0s2O12DA8Pw+l0QiaTwel00nl6pVKZ0c0ty7Iwm80YHR2lm20uQXW5XJQ4EfJWVFSUsoiSmzdOktXiPUfJOZjNZvD5fCqiSaSiSsQ6Ho8nrcoo9xwEAgE9B+65kyhKg8GA7OzsjCTQRDsHoVAIhUIBs9mMG264ASqVCjMzM7jkkktw/fXXo6WlZclmYzOAT05lksfjRbyQYrEYW7ZswZYtW+Dz+bB//37s2rULt912G9avX4+tW7fi7LPPhkAgoEaqvb291EyWKHFbW1uX5BdFMlFbWlqikhiiDFQqldTXkJxnbm4uFZxEOj9S8YzkkZgJ+P1+dHR0oLCwEFqtFsB8/BLXf7Gvry+mMCZWZjhRtTscDjQ1NWXkd0AWBnKeo6Oj1AS4v7+fnmemdpqESCb7OxAKhXRnGn49yZxl6UJ1hwkEwANQ0tsLt9+Pm266CS0tLWkTSSCxdJs1a9ZgdHQUMpkMe/fuxRVXXIH+/v6kknFO4eONt956K+a/P/XUU3jllVewb98++jtWq9XQ6/WAXg94veCzLFi/f54E/uAHERNuEiaI8aqXCSIYDOLuu+9GcXExampq0N7ejjfeeAO33XYbmpqaoNfrcejQIZSWlkK9YHaeKRDlLxnB4vP51HeVJIVl+vvCMAwlRlarFd3d3eDxeJBIJJibm6MimkzZ0RFRVkVFBcbGxnDkyBH6vOASVJLYRshbUVHRorn4RJCXl0eFmTqdDkNDQwkLWrhtdD6fH/cceDwenU2fmprCwMDAornKZOHxeGicIY/Hg0qlQmNjIw4ePIjHH38chw4dwmc/+1nk5OTg7bffhkqlyoi128cVH1mcYiwIhUJccMEFuOCCC+D3+/Hee+9h165duPPOO9Hc3IytW7fivPPOg1gsxtjYGLq6usCyLM1HlcvlGVtIuB6Va9asSZi48Hg8KBQKKBQKqt4ym80YGBigHlaECE1PT6O7u3vJPBi9Xi/a29upqi7WeRJhzNDQ0CLVI+/ddykhYr3e+Qrlxo1gWRZ9fX0IBAJobGxcEhJC0ig2b94MoVCI2dlZWCwWSi7Jeaa6K0+VSIYj/HqSOcuTSiVa+HwwLAsIhfCdfjpuvvlmrFixAj/84Q8zcs0oIViAwWBAaWlpyM9wrZO2bNmC//7v/8bk5GRCrz2FTz5ef/113HffffjHP/4RMgZ0+eWX4/rrr8f3HnwQAqEQwUAAfLEY+4JBfLOhAd3d3Th06BDWrVsHHDiA4LnnIuh2w8cwuKm6GtLzzqNepUsFgUCAffv2hbhpvPnmm/jmN7+J0tJS3HbbbdiwYQMllSUlJdBoNCk/C8KFgsQDcs2aNbTqRVrEw8PD1Ng6k+sf8T+cmZmB3+8Hn8/H9PQ0beEuhXiGz+dDq9VCrVbDZDKhra0NEokEYrEYMzMzdJOfCoGMBIlEgrq6upAIxUgzo6SVzyVvyRJphmEgl8shl8upX+Xg4GCI4jwWvF7vIjuj1atXo729Hb/61a/w7rvvYtOmTfjCF76A3//+95QvEGuhTzOWvc2dDgKBAA4cOIDdu3dj3759qKysRG9vL370ox9h69atIWopQtjSSbVhWRa9vb0IBAIJeVQm+p5cc1SGYeD1etHS0rIkqTbp5Hjb7fYQPzbt2Bi0N94YUpkMbNhA7ZFqa2uXhEiaTCaMjo6itbU1YtuF7JS5sWNKpTLhmaVMEcloYFkWJ06cQEFPD/Lb2/HTf/wDr9psKC0txUMPPYTW1taMXLdE0m2IiTHDMDh06BCuuuoqjI6OUgFOvGScJHGqtLn8iLlm19TUwOPxUJurTZs2URJ411134YknnsA6nw8/Pfdc1N10E7oXRnRuuukmPPDAA/Nk8p57wP7wh2CCwaRFNUuBYDCIffv24Wc/+xlUKhVuu+021NbWUjFLMnnc4R6+iVqYeTweWnRI1xczEoklzzLyGbjimYqKiiVJgyHOIMTGJxAIQCKRJCRESgckochgMEAqlUIikWB6ehoAaKU2k37IXFV2JL9KojcwGo00E7yoqAh9fX3YtWsX/v73v6O5uRnXXnstLrzwwiWLllwmfHLU3JlCW1sbrrnmGmzcuBGdnZ2oqqrC1q1bcfHFF0Mmk0VMtYmn5OLOA/rWrcuIojoWxsbGYDAYoFAoYLPZaLk+U18WEl+YCZJEdoXud95B7vHjwGc+A9kFF2BoaAjZ2dlLdo2MRiP0en1CqnAg1ArD4/GgsLAQSqUyqiNArAjGTIBlWXR1dSErKwvV1dUIBoO0pX3GGWfg5ZdfRllZGR544IGMHC9eus3DDz+M3/72txAIBJBIJPjlL3+JzZs3R31tmjhFJpcfmV2zF9rZ//3ss7jx0UcXVSYFEknyM5RLBJZl8fbbb2Pnzp0oLCzE9u3bUVtbS9PJopHK8Fn3dNLFvF4vnd0nc8mJFDNI98pkMtGI30SMzJ1OJ0ZGRjA3N0dDD9IpenAtdEQiEbXQIWvvzMwMRkZG4PP5aIpPptd9EqdoMpng8/kQDAbp+plutGUscOcqJRIJcnJyMDMzA7fbTRXxer0eu3btwmuvvYaqqipce+21uPTSS5dECPoR4d+PTH7zm9/ELbfcgtraWgSDQXR0dOC5557D66+/jtLSUmzduhVbtmxBXl4e/YJMTk5G/IIAi+cB2x54ALILLkBZWVnGz52Ykc/OzoYIVYhCmKvIixYpFQ+p5mwnAvJlHxoaAsMwKC0tTdj+IhmMj49jYmICzc3NKc1FEgEPERqFPySWg0h2d3fTWSSWZXHHHXfA5/Ph4Ycf/jgPYWcKp8jk8iNzazbHEsjNshh+9FHU33gjAGDi+efxyLXXYqSiAiMlJfjZz36GM888M2OHTgcsy+Kdd97Bzp07IZfLcfvtt2PlypW02qVSqVBSUkJHjzLhwhEOn88HvV4Pk8kUdYaT+CKSKmi8ufpY4FZGE23ZEsQjkNFeMzo6irm5OWg0mrQTirjt42AwuCgNZ3Z2FqOjo3C73dBqtSgqKlqStLzJyUkYjUbMzs4CAP75z39CoVBgenoar776KoqKirBt2zZs3bp1Sauzy4EoAQYMADAM8wMA/w9AAMA3WZZ9I977fWLJZDSQDOhdu3Zh7969kMvluOKKK3DJJZdAoVDQ0r3FYgmJS5T++tfUzDvI42H21lsh+vGPl+T8SLzjypUro34BiUWO2WyG3++nZu6JtG5tNhv6+vqWzIPR7/ejvb0dKpUKxcXFixTXySjYo8FgMMBsNqO5uTkj86/h7SsyxN7Q0LAkEYnEdF4gEKCmpgYsy+InP/kJbDYbHn300X8HIgmcIpMfBZJes4n6u9npxHqHA4ezs9EuleLZ1las+stfgEAAfgDG//5vqP/v/wDMr092ux0KhQJHjx7FFVdcgZMnT6YUZ7pUYFkW//jHP7Bz507k5ubiq1/9Ko4fP47GxkYA8/PDNTU1S2ra7ff7YTAYMDExgeLiYpSVlWFubo7G5+Xl5UGlUqVUBY0ELomNZYDOTYHJyspK2cTb4/HQhKLS0lKUlZUl/B6ki2QymRAIBBIqnhCxztTUVEZyuYm5vclkwuzsLI2VdDgceP7557F3717qW/qtb30Lt9xyyye9jU0RJcCAYRhmFYC/ANgAoBTAWwBqWZYNxHq/Tx2Z5IIIQ3bt2oVXXnkFUqkUV1xxBS677DIolUqqCLNYLMjr6kLTd74D+HxgRCJ49u6NaH2TDlI1I0/GCNdsNmNkZATNzc0ZnSvhnktbWxvdjXIRHhlGdtrxBFHhdkM6nQ5WqxVNTU1LYvPhdDpx/Phx5Ofnw263QygU0jGITFwzct8BQG1tLQDg7rvvhk6nw5NPPpnyZ4qXUPP000/jvvvuAwDIZDL89re/RXNzMwCgoqICOTk54PP5EAgEOHLkSKofLxmcIpPLj9TW7EjG5EDUymQ4zj777A9nKj9GcLlceOyxx/Dkk0/CYDBg9erVuO2227B582ZMTExAr9dDqVSivLx8SezkgPl1n8Q02u12yGQyVFZWLokCnHvMsbExjI2N0TlAn88XEiOYyOhXouCSZqVSCa1WG5F0RZo/VKlUSXfffD4fxsbG6PGSyeUmFWGTyYTp6WkUFBSguLgYPp8Pe/bswfPPPw+/34+rr74a11xzDYqLi2Gz2fDUU0/hm9/85nJEGi47OAEGzEJVEizL3gMADMO8AWAHy7IHYr3Hp5pMckGsa3bv3o09e/ZAKBTi8ssvx9atW6mhaNHgIFTd3ZhpbUXWOeek3GKOBJ/Ph46ODqpSSxVkYeJGdKlUKuTl5WFiYgLj4+NLlrPt8XioD2Y8MQ9XwW6z2ejiVVhYGHO8YOSxx2CsrERjY+OSVO9Ia7u+vp62KZxOJxUasSxLyXoqCQ0sy6K/vx/BYBB1dXUAgAceeAA9PT3405/+lPJClEhCzQcffID6+noUFBTgtddew44dO3Dw4EEA82SSWH0sI06RyeVHamt2tKSaSDOTmA+HIJvEoaEhnHnmmejs7MS+ffuwY8eOUPU3PUSS2d8ZgM/nw6OPPoqtW7eitLQU7733Hnbu3ImsrCxs374djY2N1PYtFglKFuHrNDd8gcyBy+VylJeXL8mmH/hQyEOSYMRiMcrLy1FSUrJkhIjMHep0OuTl5VGSTiqQPp8vaiJPOsfT6/XIyclBeXl5xHWbXAuj0QibzYa8vDzamn/llVewe/duzMzM4Morr8S2bduWJCL544zLLrsM27Ztw3/8x38wDMM8DOBfLMv+GQAYhnkcwGssy+6K9R7/NmSSC5Zlodfr8fzzz+OFF16AzWZDMBjEk08+iVWrVkUtv6d68y+VGXkwGKQtZpLFvXLlypRmbuKBKJ5ra2uTbgsTBTtRJ5LxAqVSiewHHwzJCtd/5SuQ33//shHJcHi9XvoQcLvd9CGQSKQny7IYGBiAz+dDfX09AOChhx7C4cOH8de//jUtgn/gwAHs2LEDb7wxP7oSHn8XjqmpKaxevRpjY2MATpHJfyPEX7OTyNl+4YUX8I1vfAMWiwX5+floaWnBG2+8gd27d+NHP/oRBAIB+Hw+fvKTn+Cyyy6jXogh6m8gfvb3MoJlWbz//vvYuXMnhEIhtm/fjubmZkpKFAoFysvLkyaVfr+fdpBcLlfMDlIwGITJZKKkq6KiIiMhDOFuIVKplM5h2my2qAbomYbP58PQ0BDGx8cBAMXFxUsaoUisd0ZHR8Hn81FeXk71EiTOMCcnB8XFxRCLxXjzzTexa9cu6PV6XH755bjuuuuWzI3ko0QKAQYMwzD/B+BAGJncy7Ls7ljH+rckkwTBYBD/8z//g+7ubpxzzjl4+eWX4XQ6cckll2Dr1q2oqqoKaTF7vV66q0r0S+FwONDR0YGVK1cmnKiSDFiWxeDgIJxOJ8rKymCxWJKytEgETqeTfoZMCFVI7JbFYkHOiRNo/t73wPh8CAoE8O7dC5bEumUQTqcT7e3taGhoSHiuKxAIULI+NzdHDfLlcnlEsjs4OAi3202rhb/73e+o+X661Y5kE2pINZT8PImAYxgGN910E7761a+mdT4J4tO1Mn8yEHvNTidnOwmEt71jZn9/RGBZFgcOHMDOnTvB4/Gwfft2tLS0UJInl8tRUVER87tLng8mkwlerzdktj0RYsKyLPXLlclkqKioSLpoQfxsiS2eVCqNaovHsixsNhtGRkYgEAhQWVmZsTlXEi1sMplCFNA+nw8jIyMIBoMoLy+HXC5fUtJmNpsxODgIl8sFkUiEqqoqFBQU4O2338Zzzz2Hnp4ebNmyBddddx2ampo+0QTyueeei9gJGBkZQX19Pe2OcS3AuHjqqafwyCOPYN++feS+S7nN/elr/icBl8uF4uJi3H333eDxePj2t78Ni8WCF154AbfeeitsNhu2bNmCrVu3oqWlhX5Z+vv76ZclVh7zUpuRE7Uwj8ejZuGRzMfJ7pTrU5Yo7HY7Ojs7M6oKl0gkKC8vR3l5OTyNjehgGEgPHcLc2rUQKZUompuLmXGdLFIhkgBCrJqIgMdisaC/vx/Z2dmUrAuFQgwNDcHlclFfxscffxxvvfUWnn/++Yy0zZJJqHnnnXfw+OOP47333qN/9/7776O0tBRmsxkXXHABVq5cibPOOivt8zqFjznCCWI6OdtpICT7G/MG+6Rq/lGBYRhs3rwZe/fuxcGDB7Fz504Eg0Fs374d69evh9lspvnYFRUVtB1NlMdEHKlUKlFbW5tS1Y0bxWi1WnHy5ElIJBJUVFTEfGZEI5BVVVUxiwfkGUEUykNDQwgGg6ioqKCbzWRA2vlGoxEulwtKpRLV1dWLzr2lpQV2ux2jo6MYHBykiuxMdaBIJrfZbIZYLEZlZSWysrJw77334oUXXoDf78eVV16J733ve1i/fv2nRgC5evVqPP/887jpppsW/Vt1dTXa2tqivjZagAGAlwA8wzDMLzEvwFkB4FC8c/m3JpPZ2dn49re/HfJ3SqUSX/3qV/HVr34VNpsNe/bswY9//GOMj4/joosuwhVXXIGmpiZqI0ByWsPboRaLBYODg2hpaVkSDyqSsx3J45FhGOTl5SEvLw81NTW0xTw6OgqRSEQXr3gkZ2ZmBl1dXWhqalqS9gSxSMLGjSj74hfpwjQ8PAyHwwG5XE6V4fEWuWiZ4aSqmiyRXPT+PB5NTuC27UdHR2kyRX19PRiGwR//+Ee88sor2LNnT8bywxNNqOno6MCXv/xlvPbaa9SYGgD92aKiIlx55ZU4dOjQKTL5aUekKuTZZ8//P/m7s89O+m0TaZ2F4+Mc18kwDDZt2oRXXnkFhw8fxs6dO+H1erF9+3Zs2LABZrMZR48ehUgkQjAYpASwvr4+Y2s7wzAoLCyEQqHA1NQUent7IRAIUFVVRTfx4WbmJJgjHoGMBjK2QPLGBwcHEzJAJ+s0iZYsLCxEVVVV3AKATCZDQ0MD3G43dDodRkZG0lJkEwGtyWSCQCBAcXExmpubceTIEdx77714//33ceaZZ+I3v/kNuru78fTTT+OOO+741BBJAHScKhXccsst8Hg8uOCCCwB8WL1kWfYkwzDPAugC4AdwczwlN/Bv3uZOBjMzM3j55Zfx/PPPY3h4GOeffz6uuOIKNDc303kNk8kEu90OkUgEr9eLNWvWLMlwNcnZJiq2ZOBwOKjYhMfjUWIZTnqIvVBzc/OSkGFSVRUIBFixYkXEmSLiEUksNJRKZcR50GiZ4alWJJPB6OgojfD8yU9+gvb2dvh8Pjz77LNYs2ZNxh6YiaTb6HQ6nHvuufjjH/9IjcgB0AzhnJwcOBwOXHDBBfjRj36Eiy++OCPnFgMfD7bw74UP1+w4opqUcrYTxCehzR0NLMvi6NGj2L59OywWC/x+P2688UaceeaZmJubo5XKTG0Uo4FUDv1+PyQSCex2e0bHl8LhcrkwMjKC2dnZRQboZOSHPOMKCwtRXFycVgeJxCeSVC6NRhN3rpxraM6t6hI7wLfffhtr1qzBddddh/POOy/k/Xw+35Ip9j9qhH/fRkZG0NDQgNraWuTm5ibjA/vRmZZH69mHI5qtic1mw7Zt2zAyMoKKigo8++yzSzJbmEnMzc1h79692L17N3p7e3HOOefgiiuuwJo1a/Doo4/SaESy8BCj7EwQi1g528mCpNoQk3TStnc6nbSquhRkmHiBklSDRIQtxCPSZrPRFjNp2wvuvz9ExOO7807M3nwz2tvbl8S0nUCv11MLIx6Ph927d+Oxxx7DF77wBbz++uvwer145ZVXMna8eOk2X/7yl7F7926Ul5cDALUAGhoawpVXXglgnpRef/31mUi3SQSnyOTy48M1O9Z8ZDI/kwLCH24nT57E9ddfTwU45513Hvr7+z8SAU4iuOyyy+D1erF27Vp0dXXB4XDg9ttvx6ZNm+iMY25uLm2nZhLhaThZWVnw+Xzg8Xh0/m8pq7rEO5KIrvx+PxwOB50HzXQ4RSAQoIbyBQUFKC8vD7mmPp+PGpoHAgFqaD44OIjnnnsOb7zxBurq6nDttddiy5YtS07ylxuJdALCv29p+MB+dGQymnqPi1i2Jrfddhvkcjm2b9+Oe++9F1NTU9Qr75MAl8uFN954A88++yz++c9/QqPR4M4778QZZ5wBhmGon1UmDGqJKry6ujrjqlyiYNfr9XA6ndBoNCgpKUl4kDxRkPY88dpMFtwWM0k00hgM0HzpS/SBOL1rF46JxUtKJA0GAywWC5qbm8Hj8fDSSy/h4YcfxiuvvEJFSsFg8FPVUkkBp8jk8iN0zY5XdYxWvUwR0dTfwIfZ3wKBAL/61a/w2c9+dtHrd+zYgUcffZRaj919993YsmVLyueTKtxuNyUlLMuira0NP/3pTzE7O4vbb78dmzdvpt6RxDcynQ5OOIHMycmhcYpkDSHtaI/Hg4qKCigUiiVJgSEVyNnZWQgEAni9XpSVlSWcb54qWJalo0NZWVnIycnB9PR0iJ3Q+Pg4du/ejVdffRVlZWXYtm0bLrvssiVb55cDsQpyidppxfN5TcIH9qOPU4x1srFsTerq6rB//36UlJRgYmICZ599Nnp7e5P6EB81PB4P/vM//xM1NTXYvHkzdu3ahSNHjmDz5s248sorcfrpp4PP5ycdncWdA5xbvTpjOdvRQExgV69eTc/V5XIlZY8TC8FgEJ2dndQKIxNwOp3zOdz79yO/rQ3+M87ASEkJmpqalmyBGRsbg8lkouk8r732Gh544AG8+uqrS5Km8wnGKTK5/Ii+ZidhCfRRYceOHZDJZLj11ls/snOIBpZl0d7ejp07d2Jqagq33XYbzjjjDFitVgwPDydNKknHxWQy0WeCSqWK6hZBQPK47XY7Kioq0jY/D0+B4XoXMwyzyAA9U16c4eCKeex2O1iWxbvvvovKykqYTCbs2bMHeXl52LZtGz73uc997DuYiSITdlrh/CuaD2wCz6eUb6RlEeCMjY2FzPap1WpqpkxinwCgpKQEZrN5OU4p47juuutoK/HSSy+Fz+fDO++8g127duH73/8+NmzYgK1bt+Izn/kMhEIh3YUODAxAJpPRXSi5UULmAIVCdP7852i45polI0gkdaa1tRV8Ph8lJSUoKSmhszJ6vR5zc3NJiWK4CAQC6OjogEKhgFarzdh5S6XSeWL6X/81P7je2YksoRBdXV3U5y2TyvDx8XEYjUa0tLSAz+fjrbfewn333UejO1NFvHQblmXxrW99C3v37oVUKsWTTz6JNWvWJPTaUziFqKTxtNPm/z/DM5OfRjAMg5aWFuzatQudnZ3YuXMn7r33Xtx+++0488wzYbVaceLECUilUlRWVka0+IlGIGtraxPuYkilUqxatQoulwujo6MYHh6GVqtFcXFxwutcMBgM6ZoVFBSgrKwMq1atWvQefD4fWq0WarUaRqMRx44dQ35+PsrLy9Oep+dWQufm5qiYx+l04sUXX8SBAwfw17/+FQDwwx/+EF/84hc/dd2eaCKaPXv24Nprr6Xq9JqaGhw6dChkzpjbCbjkkktoJ+Ddd98N8YF95JFHlrzQkRCZTEW9x8XHWcmXCYjFYkokCYRCIS688EJceOGF8Pv9+Oc//4ldu3bhhz/8IVpaWrB161ace+65qK2tpRYPXBuf0gULDyYQQJBlscpkArMERJJlWTp0TVq2XITb49hsNkxMTKCnpyeu7yJBIBBAe3s7ioqKoFarM/4ZgHmRSU9PD1pbW5GTkwOfz4fJyUlq2UNIMNltp4KJiQlMTExQIvmPf/wDP/3pT/Hqq6+mNXYQCARw8803h4yBXH755SHpNq+99hr6+/vR39+PgwcP4utf/zoOHjyY0GtP4RQ+KkugVPDwww/jj3/8I9atW4df/OIXH7sKFMMwaGpqwrPPPosTJ07gZz/7WQipnJqaohY/hFRyu1Jk3CkZAhkJEokEK1euhMfjwejoKEZHR+l4UqT3JTGCRqOREsiSkhLqQhEPPB4PpaWltOjT2dmZkgE68bokRFYul0Oj0SAQCODll1/G7t274XQ68bnPfQ5PPfUUysrKMDo6iieeeCIil/i0IhE7rSuvvHIR9wCAz3/+8/j/7Z15WFRl+8e/Z9iM2EF2ZYlFVglTciM3IA0BlxB9E0vNNMlE0zB/5VIqmqamvvlaVmopVyzmhr4guCKuCIoQooDsmzCsysjM8/tD57zsDDAwC8/nuuaKc+acOc+ZK5+5z/3c9/c7Y8aMXh9jU0QKJs+fP9+ji3Qka2JgYICioiJ2mVucDjHSgqKiIsaPH4/x48eDz+fj2rVriIyMxMaNG2Fvbw9fX194eHg0k/F5MHAgXBQVwRACRlkZChMnQiDmcQkdW3g8nkj2hRwOB3p6etDT02vWFJOZmdlul2FjYyNSUlJgZGTUppSNOBAKwzfV81RSUmqWXa2oqEBBQQHS09NFDoKblhkUmZujoKCADSSvXr2KtWvX4syZMzAwMOjR+G/evAkrKyu2hjQgIAAnTpxoFhCeOHECgYGBrIwJl8tFUVERcnJyOj2X0g9puaQtBkkgcdFRcmLJkiX4+uuvwTAMvv76a6xcuRK//vqrBEbZOQzDwMnJCWFhYUhLS2ODytWrV2P06NG4du0aKioqwOfzoaurCxMTkx4HkG2hoqICGxsb8Hg85Obm4saNGzA1NYWxsTE4HA64XC6Ki4vB5XKhpaUFQ0NDkQPI9u5b2ARTUVGBf/75B4qKijA3N2/XWaxlRlZLSwtGRkYYNGgQoqOjERERgdLSUvj6+mL//v2tGjPNzc2xcePGbo1XGpA3Oa226JNl7uHDhyMzMxPZ2dkwMTFBWFgYjh49CgDw8fHBTz/9hMTERNy9exdqamqorKxs9TSakZGBWbNmsdtZWVnYuHEjli9fLjVF26KgoKCAsWPHYuzYsRAIBLh16xYiIiIQGhoKKysr+Pn5wcvLC8f4fNSuXw+boiIU2digXkkJBq9qVsQlgv3PP/+AYZg2lzY6g2EYaGtrQ1tbu5VIutCHW0tLC6mpqWLpPG+PtgLJligoKGDgwIEYOHAgKz4uDILV1NTYILhpcXnTMgOipATujh1w+eADKCoq4saNG/jyyy9x6tQpsdxXR2UgHR0jrGPq7FxKP6O9JW0pWc4WNTnx8ccfw9vbu5dH03MYhoGDgwOOHDmCw4cPIzg4GDU1NRgzZgx27NgBDoeD7OxsFBYWQkVFpVcMLABAWVkZVlZWGDx4MB49eoQrV66w+pWGhoawtbUVayDbVAC9qqoK2dnZzQTQAaC6upoVVtfQ0GBtFWNjY7F582Y8fvwY3t7e2Lp1KxwcHKQ6WOoJ3UnIiaotLC30OJhsb82+sLAQCxcuRHR0NBQVFbF37154eXmxsiZCfbyQkBC4urqirq4OQ4cOxejRoxEaGtqqo9vW1pZVc+fz+TAxMWmW3g0ODpbKou2O4HA4cHNzg5ubG7Zu3YqUlBSEh4fj//7v//D6668jKCgINvPmYYimJttokpyc3GzpuTvSPQKBAGlpaSJL83RGS5H0uro6FBUVIT09HaqqquDz+eDxeGIv2ha683TFYail+HhNTQ1KS0uRnZ0NFRUVVrdM9fJltsyAEAK7khIIFBVx584dBAcH48SJE2JbshflCbS9Y2Tt6ZXSB7S3pC1ly9ltIVylAl7+tjg6OrZ7rLTVCp86dQp37tzBzz//DD09PYSGhmLOnDlYtWoVJk2aBC6Xi4yMDCgpKbEi3+KiZTe4hoYG7O3tUV9fj6KiIqiqqkJDQ6PX6g01NTVZAfTMzEykpqayvwtGRkYYPHgwLl26hF27duHevXvw9PTEmjVr4OrqKlc1kOJMbPn4+GDOnDlYsWIFCgsLkZmZiREjRohzuGKlx8Fke2v2xsbGiI6OZrenTJnS5peqq6uLAQMG4Pr16806ujuSB4qLi8Mbb7zB6unJAxwOBy4uLvjrr7/g7u7OBiw+Pj7Q09ODr68vvL29YW5uznpb379/HwDYwFIUfS1hR7VQI03cMAwDZWVlVFZWwsnJCa+//jpKS0uRkpLCisyKOtaO6E4g2dZYNTQ0oKGhwQbBwrFq6uvDSVhmoKICjBuHlJQUBAUFNdNyFAeiPIG2dwyPx5Opp1dKHyBFS9pdZfXq1UhOTgbDMDA3N8d//vOfNo+Txlrhlr+Ff/zxBzIyMvDdd99h27Zt+OKLL+Dp6Ymqqqo2HW66inBFqGnmz8DAANbW1s0CtEGDBqGwsBC3b9/utW7suro61s5QWCv68OFDzJ07F5aWlsjPz8e4ceOwePFijBo1Smr1RcVBVxNb7SXkHBwc4O/vD3t7eygqKmLfvn1S/b1JhQOOlpYWuFwuu62trY3Kysp2j58/fz5cXV0RFBQE4OXTwO+//w4NDQ2pLdoWlejoaLz77rvsZEAIQUZGBiIiInD69Gmoq6vDx8cHU6dOxcCBA5v5xAoEAlaPq60uO2EjTHecc0SloaEBycnJsLKyambnB7zUbisrK0NpaSn4fD4rkt5Vq0ZxBJKdUVRUhLKTJ2GQno5fHj9Gsbk5rly5gpMnT2LIkCFivZYo7jZnzpzB3r17WR/hZcuW4ebNmyKdK0FoirTveTln94KrjTTRkdycNPLw4UNs2rQJDx8+xBdffAEvLy9UV1cjKysLCgoKsLCwEMmlq6knd3l5eZt6lO0hEAhQVFSEvLw86OjowMzMrEemFM+ePWMDSCUlJRgaGkJXVxd3795FREQELl26hLfeegvPnz9Heno6Nm3aJBNlCz1BmqWtRETyOpOd0VEB6rx580QOJnk8HoyNjfHgwQO28aGkpIT1E/36669RVFQktUXbPYEQgqysLERGRuLvv/+GiooKpk6dCl9fXxgaGrJOAaWlpWhsbGwWrL148QIpKSkwNjbutczV8+fPkZycDBsbm05lCIQi6aWlpeDxeCLL+PRFIFlRUYHMzEy8+eabUFZWRmJiIkJCQqCpqYmKigpMmzYNX3/9tViv2Zm7DSEEQUFBOHfuHFRVVfHbb7+xmmJtnSsl0GCy7+kX7a4RERE4d+4cfvnlFwDAkSNHcOPGDezdu1fCI+uYR48eYdOmTUhPT8fKlSsxefJkVFdXIzs7GwzDwNLSslVQKTRqKC4uRnl5eZtScl2BEILi4mLk5uZCU1OzSxI/DQ0NrJ0hh8OBoaEh9PT02IRHTEwMnJycEBAQAE9PTzZYraqqwtOnT7tlVCFLyEFiS/qDyY7oinD5iRMnsG/fPsTExLT5fk5ODry9vZGamtqbQ5Y4hBDk5uYiKioKx48fByEE3t7e8PPzg6mpKRobG1FWVoaSkhI0NDSAx+PBzMwMgwcP7pWaumfPniElJaVbouqNjY0oLy9HaWkp6uvrWdHcliLpwkDS2dm5y9lMUeFyuazEkIqKCjIzMzF37lz88ccfcHZ2Rn19Pe7fvw83N7deub6cQYPJvqdfBJNCG72mweTNmzexZ88eCY9MNB4/fozNmzcjNTUVK1aswHvvvYeamhpkZWUBACwsLKCgoMBmIFVVVdnMn7iWOgkhrDWkmpoazM3N29TGFK5+lZSUgBACAwMDDBw4EE+ePEFERASio6NhYWHButG09RnyREeJsbffflvWE1uyHUyuWrUKurq6rKViRUUFtm3b1uaxAQEB8PLywkcffcTuExZtV1RUYOTIkSgsLMTbb7/drs+3ubk51NXVoaCgwPoXA7LpEw68nBSKiooQFRWFqKgoPHv2DN7e3qzkwPHjx+Hj44P6+no8e/aMzQKKy2O1vr4e9+7dg52dXbvSEKIiFEkvLS1t5m2upKSE1NTUPgkkXVxcMGDAAOTk5GD27Nn47bffWIFwWcHDwwO1tbVITExstv/+/ftwdXXFoUOHMGfOnN4eBg0m+5rERCLPy9tCZG2Zuz2ysrIQGhqKu3fvYsWKFTA2NsajR49gaGgIDoeDQYMGwczMrFdr5QghrIvPgAEDYGFhARUVFZSVlaG4uBiNjY3Q19eHgYEBSktLERkZiRMnTmDgwIGYNWsW/Pz8ejzvyyMymtiS7WDy6dOn8Pf3R25uLgYPHozw8HDo6Og06wgHwHpGZ2VlNfufd+7cuUhOTkZxcTH09PQQHx+PQ4cOtevzbW5ujtu3b7cSmpZ1n3Dgf0+bx48fZwvAZ86ciY8//hg2NjYQCARsFlBoBN8TMW+hNE9v+GALXRry8/NRXl4OPT09GBsbi1Qf1FWqqqqQnp7OBpJ5eXmYNWsWDhw4IJYOOlEeVPLy8hAYGIji4mJwOBwsWrQIn3/+OYCudwlu2LABmzdvRnV1NbvURAiBu7s7OBwOLl261ON7EgEaTPY1qqpEWqwRexMprxXuEg8fPsT+/fvxxx9/QE9PD9OnT8fKlSvB4/GQlZUFPp8PS0vLXrPRFdLY2IicnBwUFBRAIBDg9ddfh4ODA2pqapqVVvn7+2PmzJk9MmqQNsSlDNBUjWDnzp24ceMG6+AjI8h2MCkuRF0uby+YlAefcCHp6emYPXs2fvjhB+Tk5CAqKgrFxcXw8vKCn58f7OzsmjkRCLOABgYGHVolNhXyrnZw6PX6xZqaGqSmpsLJyQmNjY2s9EV7+pDdobq6GmlpaRg6dChee+01FBYW4v3338eePXswZswYsdyHKA8qQocdV1dX1NTUYNiwYfj7779hb2/f5cLu8+fPw8PDA4mJiayLwqFDh7Bw4UIkJSXByclJLPfVCTSY7GsUFQn4fEBBAfj2W0DGMnVdoTu1wu2tSkmSgwcP4rXXXsPUqVNRWVmJzZs34/bt21i+fDl8fX1RX1+PrKwsNDY2wtLSUqyrZcLkQklJCerq6lhNyqdPn+LDDz9ESUkJNDQ0MH/+fPj7+8PIyEjupMf4fD5sbGyaKQMcO3asW8oAwsRWUzWC3tJY7iVoMAmI3hVuYWEBbW1tMAyDTz75BIsWLerS+bLA/fv3oaCg0OwfBJfLxalTpxAVFYWcnBxMmjQJ06ZNg7OzM4CX2bPS0lJUVVVBS0sL+vr60NbWZrOALf3Cb2/bBvOAgF5bdhYGki2XtpvqQ5aXl2PAgAGsPqSSklKXr/HgwQM4OztDVVUVxcXFmDlzJn744QeME6OkSnceVHx9fREUFAQPD48uB5O1tbXQ0tLC9u3bsXz5cnC5XNja2mLOnDnYuXOnOG5JFOTrV0cW6CeZye7SXiJB2sjLy8OWLVtY5Ybp06fj2bNnyMrKwosXL5r9hnUVoS1uSUkJqquroaurC0NDQxBCcPr0aURGRqKyshLTpk2Dg4MDDh8+DB0dnXZlmmQdeSmZEBPdnrP7xAFHnHRU/CoqCQkJMDY2RmlpKTw8PDBkyBC4u7uLc5gSp63Mk5aWFubOnYu5c+eipqYG0dHR2LlzJx4+fIgJEybAz88Pw4YNAwDW+urhw4esfpnBpUut/cL7OJAEWutD1tbWoqysDHfv3oWioiIbWHYme1FbW8teQ1VVFaWlpXj//fexdetWsQaSwEvFAeETqtDbtiNycnJw9+7dZo0+XfEsVlNTw9ChQ3H9+nUAwNq1a8HhcLBhwwYx3A1FapESlxtKzxg0aBD+/e9/Iz8/H6Ghodi9ezcbVD5//hzZ2dnIysqChYUFdHR0Og0qhb7cJSUl4HK50NbWhomJCczNzRETE4P169fjyZMn8PHxwc6dOzFkyBD2M6dOnYr6+vq+uG2JQF3ExIPMBZMd2RKJ6vMtlMbR19fHtGnTcPPmTbi7u/cLn3Ah6urqmDVrFmbNmoVnz57h7NmzOHDgAO7fvw93d3f4+fnBzc0NHA6HdVYo0dWFay/7hQP/CySHDh0qUmegmpoa1NTUYGFhwQq637t3DwzDsPJILaUv6urqmnWGC+t2v/32W3h4eHRr3OJ40AFeBrkzZszArl27WJmQ7ngWjx49GidPnkRSUhL279+PQ4cOiaRlR5FhZMDlRpIwDANPT89Wq1LSiqmpKfbu3YuCggKEhobixx9/RFBQEGbOnImGhgY2qLS0tGwVVApFzYuLi1FRUQFNTU0YGhrCwsIC8fHx2LZtG9LS0jB58mSsX78ezs7O7daiy3OHNnUREw9ytcwtSld4XV0dBAIB1NXVUVdXBw8PD3zzzTd499132fMXLVoENzc3lJeX46233ur1ZglpoqGhAbGxsQgPD0dSUhJGjRqFadOmYdSoUYiLi4NWejpsCgtRaGsL/vDhMDAwgJ6enti6Daurq/HgwQORA8nO7kWoZdlUd5NhGKSkpLC1nlwuF9OnT8dXX30FHx8fsdxHS0Rd5n7x4gW8vb3h5eWFFStWtPlZonYJ/vXXX5g1axYcHBygo6ODy5cvi+VeugCdkfsemZmzJaE4UFhY2GxVas+ePTK1KlVYWIitW7fi8uXLWLp0Kfz9/dHQ0ICcnBzU19fD3NwcKioqzUTNDQ0NoaGhgatXryIiIgK3b9/GhAkTMHv2bDZhIOv0pBaWLnM3g9ZMAqJ1hWdlZbGWV42NjZgzZw5buC08PykpCTo6Orh16xYOHDjQ680S0gqPx8OFCxcQERGBuLg4CAQCrF+/Hn5+flBSUkJtbS07ab322mvs8nJ3G2KEjTDCZWdx8uLFC5SVlaGoqAhcLhdGRkaoq6uDtbU1/P39ERwcjBkzZoj1mk0R5UGHEIJ58+ZBR0cHu3btavZed7oECwoKYGpqCgUFBSQlJbG1sX0IDSb7HpmZsyWtOCDL83RRURG2bduGCxcu4NNPP4WtrS3S0tJYm9fnz59j3LhxSEpKQkREBK5evYrRo0cjICAA7u7uUm3L1x16UgsrT8oAYqD7czYhpKNXv8TGxoYUFhYSQggpLCwkNjY2nZ7j4+NDYmJiCCGErFu3jnz//fe9Osa+5NSpU2TEiBEkMjKSLFmyhDg6OpK5c+eSiIgIUlFRQerq6khJSQm5d+8eiY+PJ1evXiWZmZmEy+WSuro6kV5FRUUkLi6OlJWViXxOV1/l5eUkLi6O5Ofnk/T0dDJx4kRiYGBA3nvvPZKQkED4fH6vfYfl5eVkwoQJxMrKikyYMIE8ffqUEEJIQUEBmTx5MiGEkCtXrhAAxMnJiQwdOpQMHTqUnDlzhhBCyAcffEAcHR2Jk5MTmTp1Kvv/Z0dwuVyirKxMPv/88167r07obH6hL/G/ZIbY2FgCgCQmJrL7fv/9d6KoqEju3bsn9uvV1taS6upq9u+RI0eSs2fPiv06fUV2djZZs2YNMTAwIDY2NmT58uWkvLycnD9/nowePZro6uqSd955h5w+fZrweDxJD7dXMTMzI2VlZd0+/8yZM8Ta2ppYWlqS7777Towjkzm6PffIVWZSXHS1qzsnJwfu7u5ITU2FhoaGPFgqNePAgQOYOXMma5HI5/ORkJCAyMhIXLhwAfb29vDz88OkSZOgqqqKuro6lJaWoqysjG2I0dfXh7KycpufX1VVhbS0NLi4uIhs69VVhFaPQmH1uro6zJo1C3PmzIGRkRGioqJgZmaGb775pleuLwlWrlyJo0eP4p9//pGUqDDNTPY9MjNn97XiQEerUk2ZP38+Tp8+DX19fbaURBoNLY4ePQoejwc/Pz80NDRg+/bt+OWXXzBx4kTMnj0bjo6O2LVrF5SVlbF7926JjrW3aU+hhdJl6DJ3VxGXV3htbS3eeecdrF27FtOnTwfQf7zCgZcyE7du3UJ4eDhiY2NhZWWFadOmwdPTE2pqamxDTGlpKRiGYQPLAQMGAPifWLhQ47E3EAaSQqvHZ8+eISAgALNmzcLChQt75ZqSor6+HikpKbhy5Qq++uorhIeHsz+gEoAGk32PTM3Zw4YNg7W1NcLCwrB06VJERUUhIyNDoo1ily9fhpqaGgIDA9lgUlYMLerr61uVCAkEArmoi+wIWa+FlSJoMClOJNEsIQ8IBAIkJycjPDwc586dw+DBg+Hr64vJkydDU1MTz58/ZwNLQgjU1dVRUVGBN998s9cCyYaGBty9exe2trbQ1tZGQ0MD5syZg6lTp2LJkiVy17V38uRJ+Pr6wsTEBGvWrMHSpUslORz5+nJlA5mas5ctW4aTJ08iKioKw4cPx6FDh/DBBx9Ielit5m15MrSQd2S5FlYK6PacLd+PK93Ex8cHhw4dAvDSNUTocd0UQggWLFgAOzu7VoHk0aNHYWtrCysrKwQFBcHR0bHVucuWLYOVlRWcnZ2RlJTEvnfu3Dn23NDQ0F64u96Dw+HA1dUVW7ZsQVJSEr777jvk5ORg6tSpmDlzJsLDw6Gmpoa33noLwMsMrpKSEu7fv4/s7Gyxa5nxeDwkJyfDxsYG2tra4PF4mDdvHry8vMQWSFZUVMDDwwPW1tbw8PBoN4Ntbm4OJycnuLi4sPfflfNFxcfHB4QQ5OfnSzqQpFA6ZcyYMXjy5AkCAwMxevRoqQgk26KrOrGUjpk/fz709fWb/TZ2dy6sq6tDTU0N+3dMTEyr31xKH9BJUWW/pCfNEo2NjURNTY3Y2NgQBwcHoqGhQS5evNjs88+cOUPeffddIhAISGJiIhkxYgQhhJDGxkZiaWlJHj9+TBoaGoizszN58OBB3958LyAQCEhaWhrZuHEjGTFiBBk1ahQxNzcnCQkJpLa2llRWVpKHDx+Sq1evkvj4eHL//n1SUlLSo2abyspKcuHCBZKbm0vq6uoIl8sl06dPJ1u3biUCgUBs97Zq1SqyZcsWQgghW7ZsIatXr27zuPYKxEU9X0aRdDNKf3zJFPn5+QQAUVBQICkpKZIeDkt2djZxcHBgtzU1NZu9r6Wl1ccjki8uXbpE7ty50+w77u5c+PjxY+Ls7EycnZ2Jvb19rzXQZGZmEkVFRfLNN98027948WKipqZGbt261SvX7WO6PffI1cQkDVy7do14enqy25s3byabN29udsyiRYvI0aNH2W1h97go58o6165dI/b29iQkJISMHDmSjB8/nvzwww/k8ePHpLa2lnC5XPLo0SNy7do1EhcXR1JSUkhxcTGpra3tMHh8Fh9PGtavJ8/i4wmXyyUXLlwgT548IXV1daSqqor4+/uTb7/9VqyBJCGid/63F0x2RzlAhpB0YNUfXzKFFCgOtEnLYFLO/51KBFn8jhcvXkzU1dXZuXzDhg1EWVmZxMbGSnhkYqPbc4/MOeBIO6JYM7V1TEFBQb+wdQoLC8O5c+cwaNAgEEKQm5uLyMhIfPjhhwAAb29v+Pn5wdnZGXw+H0+fPmWXwHV1daGvrw8NDY1mS9TNPMOVlZG+fTss3nsPenp64PP5WLp0KWxsbLB27Vqx10iKuvzVnvMGXT6j9Gc2btwIHR0dqbf5FJY+hYSEtFv6ROkZsjAXrlu3DocPH8bWrVsxZMgQbNiwAceOHcOkSZMkPTSJQ4NJMUNI59ZM7R0jyrmyTlOJCoZhYGZmhhUrViA4OBhFRUWIjIzE4sWL8fz5c3h7e8PX1xdOTk4QCAR4+vQp8vLyUFNTAx0dHRgYGEBTUxOcy5f/5xne0ACz7GwMGDgQAoEAy5cvh7GxMTZs2NDt75L6wVMo4qOp4sDu3bsRHh4uKemqNpk9ezYuXryI8vJymJqaYsOGDQgJCYG/vz8OHjzIGmK0pC1JIVl2Q6O0xtDQEMuXL8eOHTvQ2NiIH3/8Ef7+/pIellRAg0kxY2pqiry8PHY7Pz+f9QLv7Bgej9fpufIKwzAwNjbGZ599hqCgIJSWluL48eMIDg5GVVUVpkyZAl9fXzg4OIAQgoqKChQUFCA9PR2mZmawVlJ6GYwrK0N1yhQ0CgT44osvoK6uji1btvRIGoP6wVMo4uP8+fOs4sDu3bslKV3VJseOHWtzf1xcXIfnffjhhwgKCkJgYGCz/cHBwbSzWARkZS60trZGQ0MDxowZQ5scm0C7ucXM8OHDkZmZiezsbPB4PISFhbXye/bx8cHhw4dBCMH169ehqakJIyOjVuf+/PPPOHDgQLud3X/++SecnZ3h7OyMUaNGISUlhX2vve5hWYBhGBgYGGDx4sWIiYlBdHQ0TE1NsXbtWrzzzjsIDQ1FaWkp7O3t4ebmBsWxY3Fj0yY8CgzEqc8/R3h+PkJCQsDhcPDDDz/0qsaaKJ3/HXUbinI+hSJPyKvigLu7O2vsIM+01Ym9fv16mJiYwMXFBS4uLoiOju7y58rCXBgfH49PPvkEI0eOREJCQrPf3H5PJ0WVlG7QljXTTz/9RH766SdCyMvu5k8//ZRYWloSR0fHZl1gwnMtLCyItrZ2h53dCQkJpKKighBCSHR0NNsVTkjP7aWklcrKSnLo0CHi6+tL3nzzTRIcHEyGDRtGwsPDSW1tLTl37hwZN24c0dXVJYGBgeTUqVOkoaGh18YjSud/R92G7Z0vJ0i6GaU/vigSpGVTybp164iZmRlxcnIiH330ETtfyzJtdWJ31UI4ICCAGBoaEkVFRWJiYkJ++eUXqZ8L79y5Q9TV1cmnn35KGhoaiJmZGZkyZYqkhyVuuj33UNFyKSUxMRHr16/Hf//7XwDAli1bAABr1qxp8/jKyko4OjqioKAAQM+M72WF8vJyeHh4QE1NDTU1NZgwYQK4XC4aGxtx8OBB3L59G5GRkfjyyy/ZmiVKnyJfBb+yAZ2zJUhLsXN5dUNreZ/yLhT+6NEjjB49GmPHjsVff/0FDoeD3377DfPnz8elS5fkqf6dipbLG+11fLfHwYMHMXnyZHZb2D08bNgwHDhwoFfHKilCQ0OxaNEiXLlyBdeuXcOIESOQm5uLX3/9FUpKShg5ciS2b99OA0kKhSIRDAwMoKCgAA6Hg48//hg3b96U9JB6jb1798LZ2Rnz58/vsfmCNFFcXAxPT0/Y2dnhzz//ZMumAgMDMWTIEISEhEh4hNIBDSallLYyxu11I1+4cAEHDx5s5hWbkJCApKQknD17Fvv27cPly5d7baySYtOmTViyZAkAQFVVFQEBATh//jwUFXveVyaKG0NGRgZbI+Ti4gINDQ3s2rULgHhqiCgUimxTVFTE/n38+HG5dWZZsmQJHj9+jOTkZBgZGWHlypWSHpLYMDQ0RFZWFi5evAgVFRV2v4KCAtLT03Ht2jUJjk56oMGklCJKVzgA3Lt3DwsXLsSJEyegq6vL7m+re1jeaPoPW9yEhoZi4sSJyMzMxMSJE9tsgLK1tUVycjKSk5Nx584dqKqqNutMDQ4OZt+nciAUinwze/ZsjBw5EhkZGTA1NcXBgwexevVqODk5wdnZGRcuXMDOnTtbnZeXl4fx48fDzs4ODg4OrHyauK1We5P+lIGltA0NJqUUUbrCc3NzMX36dBw5cgQ2Njbs/ra6h58/f96h5/fFixehqanJZtI2btzIvifLfuHd5cSJE5g3bx4AYN68efj77787PD4uLg5vvPEGzMzM+mB0FApF2jh27BiKiorw4sUL5OfnY8GCBThy5Aju37+Pe/fu4eTJk6wod1MUFRWxY8cOpKen4/r169i3bx/S0tJEeqDtCr0ZtPaXDCylAzrp0KFIkM66whcsWEC0tLRYb/Bhw4YRQlp3D2/cuLFTz+8LFy6Q9957r9UY5NUvvDO66sX70UcfkT179rDb8tjF2Q0k3dncH18UGcfHx4fExMSI3V6wsLCQ3LlzhxBCSHV1NbG2tiYPHjzosid2W53YH3zwAXF0dCROTk5k6tSp7LgpMgft5qa0jyid4RcvXsT27dtx+vTpLp8rq3TkbDNv3jxwuVx2n7a2drtP7DweD8bGxnjw4AEMDAwAyG8XZxeh3dx9D52zZZicnBy4u7sjNTUVgwcPFnkO6g6+vr4ICgpCUFAQLl68yIqFjxs3DhkZGWK7DkWmoN3ckubRo0dQUlLCunXrmu1fsmQJ1NXVcfv2bQmNTPTO8MTERAwdOhSTJ0/GgwcPunSuLHL+/Hmkpqa2evn6+rJuDAA6dWM4e/YsXF1d2UASoDVEFAqla9TW1mLGjBnYtWsXNDQ0evVaOTk5uHv3Ltzc3GTCE5si/dBgUkxYWVlh4cKF2LlzJ8rLywEAGzduxK+//orjx49L1IWmrexzy85wV1dXPHnyBCkpKfjss8/g5+cn8rnySFfcGI4dO4bZs2c320driCgUiqi8ePECM2bMwL/+9S9Mnz4dALr0QNsV+jJopfQfaDApRtatWwc+n4+tW7fi4MGD2LBhA44cOYJJkyZJdFyidIZraGhATU0NADBlyhS8ePEC5eXlIneVyxshISGIjY2FtbU1YmNjWS2xwsLCZp3Z9fX1iI2NZX8AhIjSxUmhUCiEECxYsAB2dnZYsWIFu7837AX7Mmil9C9ozaSYWbt2LXbs2IHGxkbs3r1bKrxnGxsbYWNjg7i4OJiYmGD48OE4evQoHBwc2GOKi4thYGAAhmFw8+ZNzJw5E0+ePAGfz+/0XAqlHeQ/hS190Dlbxrh69SrGjh0LJycnVhB78+bNcHNzg7+/P3JzczF48GCEh4f3yPubEIJ58+ZBR0eH1cMFgFWrVkFXVxchISEIDQ1FRUUFtm3b1tPbosgm3Z6ze67uTGmGtbU1GhoaMGbMGKkIJIGX0hN79+6Fl5cX+Hw+5s+fDwcHB+zfvx8AsHjxYkREROCnn36CoqIiXnvtNYSFhYFhmDbPzcvLw/Tp08Hn87Fw4cJWDgDff/89/vzzTwAvA9n09HSUlZVBR0cH5ubmUFdXh4KCAhQVFSVaS0qhUCiSZsyYMW2WEwEvJcfERUJCAo4cOQInJye4uLgAeBm0hoSEwN/fHwcPHmSDVgqlq9DMpBiJj4/H5MmTMWzYMFy/fh13797F0KFDJT0ssSLMVMbGxsLU1BTDhw/HsWPHYG9v3+bxp06dws6dOxEfHw+gf3iGU1hoZrLvoXM2hULpLrSbW9IkJSXBz88PCxcuxMWLFzF48GB89dVXkh6W2Ll58yasrKxgaWkJZWVlBAQE4MSJE+0e31ZziiQJDw+Hg4MDOBxOh1nR9oTaZcmVgkKhUCiUvoAGk2Lg0aNHmDx5Mjw9PbFnzx4oKytj3bp1iI6OljtP7K5IBdXX1+PcuXOYMWMGu49hGHh6emLYsGE4cOBAr4+3JY6OjoiKioK7u3u7x/D5fCxduhRnz55FWloajh07hrS0NACi2SxSKBQKhdKfoMFkDykuLoanpyfs7Ozw559/sgXUgYGBGDJkSKt6QlmnK1JBp06dwujRo5sVjSckJCApKQlnz57Fvn37+jzYtrOzg62tbYfHdJR97arNIoVCoVAo8g5twOkhhoaGyMrKarVfQUEB6enpEhhR79IVqaCwsLBWS9zCY/X19TFt2jTcvHmzwyyhJGgr+3rjxg0AoAK/FAqFQqG0gGYmKV1i+PDhyMzMRHZ2Nng8HsLCwuDj49PquKqqKly6dKmZNlpdXR1qamrYv2NiYhATEwN9ff12Rb0JIVi2bBmsrKzg7OyMpKQk9r326honTZoER0fHVq+OajtbXrMl/UGonUKhUCiU7tBZNzeF0gqGYaYA2AVAAcCvhJBNDMMsBgBCyP5Xx3wI4F1CSECT8ywBHH+1qQjgKIArAGoBHCaEtIooX13rMwBTALgB2E0IcWMYRgHAQwAeAPIB3AIwmxCSJuI9XATwBSGkVRcOwzAjAawnhHi92l7z6t62MAyTAWAcIaSIYRgjABcJIR2vm1MoFAqFIsfQZW5KlyGERAOIbrFvf4vt3wH83mJfFoBWWkkMw5h3cDlfvAw0CYDrDMNovQrizAE8evWZYBgm7NWxIgWTnXALgDXDMBYACgAEAJjz6r2TAOYBCH31X9HSnRQKhUK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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "# let's plot the transformation\n", "\n", "A = np.array([[-2, -4, -20], [2 , 6 , 24], [2, 10, 32]])\n", "\n", "# construct some points in 3D space\n", "x = np.linspace(-1, 1, 5)\n", "y = np.linspace(-1, 1, 5)\n", "z = np.linspace(-1, 1, 5)\n", "# this creates a mesh of points in 2D\n", "xx, yy, zz = np.meshgrid(x, y, z)\n", "# convert to row vectors\n", "xxx = np.reshape(xx,(1,np.size(xx)))\n", "yyy = np.reshape(yy,(1,np.size(yy)))\n", "zzz = np.reshape(zz,(1,np.size(zz)))\n", "# convert to a 3 x N matrix of vectors/points\n", "vecs = np.vstack((xxx,yyy,zzz))\n", "\n", "# transform these points\n", "Avecs = A@vecs\n", "\n", "# plot\n", "fig = plt.figure(figsize=(12, 5))\n", "\n", "ax1 = fig.add_subplot(1, 2, 1, projection='3d')\n", "ax1.plot(vecs[0,:], vecs[1,:], vecs[2,:], 'r.')\n", "ax1.set_xlabel('$x$', fontsize = 16)\n", "ax1.set_ylabel('$y$', fontsize = 16)\n", "ax1.set_zlabel('$z$', fontsize = 16)\n", "ax1.set_title('$\\mathbb{R}^n$', fontsize = 16)\n", "ax1.set_xlim3d(-1, 1)\n", "ax1.set_ylim3d(-1, 1)\n", "ax1.set_zlim3d(-1, 1)\n", " \n", "ax2 = fig.add_subplot(1, 2, 2, projection='3d')\n", "ax2.plot(Avecs[0,:], Avecs[1,:], Avecs[2,:], 'r.')\n", "ax2.set_xlabel('$x$', fontsize = 16)\n", "ax2.set_ylabel('$y$', fontsize = 16)\n", "ax2.set_zlabel('$z$', fontsize = 16)\n", "ax2.set_title('$\\mathbb{R}^m$', fontsize = 16)\n", "ax2.set_xlim3d(-20, 20)\n", "ax2.set_ylim3d(-20, 20)\n", "ax2.set_zlim3d(-20, 20)\n", "\n", "# rotate to try and get a better view - with a different plotting backend you could rotate with mouse\n", "# you could edit this to try and get a better idea of the 3D view\n", "ax2.view_init(20, 50)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's see how we can compute the null space. \n", "\n", "Recall we said in the previous lecture that as row operations will never change the zero vector, we don't actually need to perform row operations on the augmented matrix $[A|\\boldsymbol{b}]$ when $\\boldsymbol{b}=\\boldsymbol{0}$. So we can make use of the reduced row echelon form of $A$ we created above (call it $A_{\\text{RREF}}$). The solutions to $A\\boldsymbol{b}=\\boldsymbol{0}$ will be the solutions to $A_{\\text{RREF}}\\boldsymbol{b}=\\boldsymbol{0}$.\n", "\n", "\n", "$$\n", "\\begin{pmatrix}\n", "1 & 0 & 6 \\\\\n", "0 & 1 & 2 \\\\\n", "0 & 0 & 0\n", "\\end{pmatrix}\n", "\\begin{pmatrix}\n", "x \\\\\n", "y \\\\\n", "z\n", "\\end{pmatrix}\n", "=\n", "\\begin{pmatrix}\n", "0 \\\\\n", "0 \\\\\n", "0\n", "\\end{pmatrix}\n", "\\iff \n", "\\left\\{\n", "\\begin{align}\n", "1x + 0y + 6z &= 0\\\\\n", "0x + 1y + 2z &= 0\\\\\n", "0x + 0y + 0z &= 0\n", "\\end{align}\n", "\\right.\n", "$$\n", "\n", "from which we can conclude that we actually have here two equations for three unknowns. \n", "\n", "Let's set $x = \\alpha$, with $\\alpha$ an arbitrary scalar. \n", "\n", "Then the first equation tells us that $z = -\\alpha / 6$, and then the second that $y = \\alpha / 3$. This will be a solution for any $\\alpha$, and so in this example the dimension of the null space is 1 (and the dimension of the null space plus the rank is $n=3$ as expected).\n", "\n", "For example, with one choice of the arbitrary $\\alpha$ we get one vector from the null-space:\n", "\n", "$$\n", "\\boldsymbol{x}\n", "= \n", "\\begin{pmatrix}\n", "6 \\\\\n", "2 \\\\\n", "-1\n", "\\end{pmatrix}\n", "$$\n", "\n", "\n", "Can we do this in `numpy`, and check our solution against?" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[-0.93704257]\n", " [-0.31234752]\n", " [ 0.15617376]]\n", "[[ 2.22044605e-15]\n", " [-1.33226763e-15]\n", " [-8.88178420e-16]]\n", "[[-6.]\n", " [-2.]\n", " [ 1.]]\n" ] } ], "source": [ "A = np.array([[-2, -4, -20], [2 , 6 , 24], [2, 10, 32]])\n", "null_vecs = sl.null_space(A)\n", "print(null_vecs)\n", "\n", "# check that A@ these vectors yields the zero vector\n", "print(A@null_vecs)\n", "\n", "# is this the same as we obtained above - let's normalise it and multiply it by the length of the vector above\n", "print((sl.null_space(A) / sl.norm(sl.null_space(A))) * sl.norm(np.array([6,2,-1])) )" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Of course we can always multiply the vectors in the null space by -1 (or indeed any constant) and it is still a member of the null space." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "celltoolbar": "Slideshow", "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.13" }, "latex_envs": { "LaTeX_envs_menu_present": true, "autoclose": true, "autocomplete": true, "bibliofile": "biblio.bib", "cite_by": "apalike", "current_citInitial": 1, "eqLabelWithNumbers": true, "eqNumInitial": 1, "hotkeys": { "equation": "Ctrl-E", "itemize": "Ctrl-I" }, "labels_anchors": false, "latex_user_defs": false, "report_style_numbering": false, "user_envs_cfg": true }, "toc": { "base_numbering": 1, "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": false, "title_cell": "Table of Contents", "title_sidebar": "Contents", "toc_cell": true, "toc_position": {}, "toc_section_display": true, "toc_window_display": false } }, "nbformat": 4, "nbformat_minor": 1 }