{ "cells": [ { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "# Local and Collective Noise in the Driven-Dissipative Dynamics of Large Two-Level System Ensembles \n", "\n", "Notebook Author: Nathan Shammah (nathan.shammah@gmail.com) \n", "PIQS code: Nathan Shammah (nathan.shammah@gmail.com), Shahnawaz Ahmed (shahnawaz.ahmed95@gmail.com)\n", "\n", "### Module\n", "\n", "Below we give an introduction to PIQS, the Permutational Invariant Quantum Solver [1], which is integrated in QuTiP from version 4.3 [2] and can be imported as a module simply with $\\texttt{qutip.piqs}$. \n", "\n", "![title](images/piqs.png)\n", "\n", "### Introduction to the Open Dynamics\n", "\n", "We are interested in considering the non-unitary dynamics of a quantum system interacting with a larger environment, whose evolution is goverened by \n", "\\begin{eqnarray}\n", "\\dot{\\rho} = \\mathcal{D}(\\rho) &=& \n", "-\\frac{i}{\\hbar}\\lbrack H,\\rho \\rbrack\n", "+\\frac{\\gamma}{2}\\mathcal{L}_{A}[\\rho],\n", "\\end{eqnarray}\n", "\n", "where $\\rho$ is the density matrix of the system, $\\mathcal{D}$ is the Liouvillian superoperator, $H$ is the Hamiltonian of the system and $\\mathcal{L}_{A}[O]=2AOA^\\dagger-A^\\dagger A O-OA^\\dagger A$ is the Lindbladian of the system, with $A$ a jump operator, and $\\gamma$ is the dissipation rate. Considering this Lindblad dynamics implies making the Born, Markov, and Secular approximations regarding the system-environment interaction. \n", "\n", "For a system comprising several two-level systems (TLSs), the Hilbert space grows like $2^N$ and in general the Liouvillian space grows like $2^N\\times2^N=4^N$. This fact strongly limits the direct use of numerical methods to solve the dynamics in the uncoupled basis. \n", "\n", "If additional symmetries are present in the system, the complexity of the problem can be reduced. If only collective processes are present, the collective system can be mapped to an effective large spin of dimension $(N+1)$.\n", "\n", "If also local processes are present in the Lindblad dissipation terms, in general such reduction is not possible. However, if one can assume that the local dissipation noise affects with an identical rate each of the TLSs, a reduction in the complexity of the problem can be performed. \n", "\n", "The *Permutational Invariant Quantum Solver* (PIQS) is an open-source Python solver to study the exact Lindbladian dynamics of open quantum systems consisting of identical qubits. PIQS is integrated in QuTiP and can be imported as as a model simply by importing it as $\\texttt{qutip.piqs}$.\n", "\n", "Using this library, the Liouvillian of an ensemble of $N$ qubits, or two-level systems (TLSs), $\\mathcal{D}_{TLS}(\\rho)$, can be built using only polynomial – instead of exponential – resources. This has many applications for the study of realistic quantum optics models of many TLSs and in general as a tool in cavity QED [1].\n", "\n", "We can consider a system evolving according to the equation\n", "\n", "\\begin{eqnarray}\n", "\\dot{\\rho} = \\mathcal{D}_\\text{TLS}(\\rho) &=& \n", "-\\frac{i}{\\hbar}\\lbrack H,\\rho \\rbrack\n", "+\\frac{\\gamma_\\text{CE}}{2}\\mathcal{L}_{J_{-}}[\\rho]\n", "+\\frac{\\gamma_\\text{CD}}{2}\\mathcal{L}_{J_{z}}[\\rho]\n", "+\\frac{\\gamma_\\text{CP}}{2}\\mathcal{L}_{J_{+}}[\\rho]\\nonumber\\\\\n", "&&+\\sum_{n=1}^{N}\\left(\n", "\\frac{\\gamma_\\text{E}}{2}\\mathcal{L}_{J_{-,n}}[\\rho]\n", "+\\frac{\\gamma_\\text{D}}{2}\\mathcal{L}_{J_{z,n}}[\\rho]\n", "+\\frac{\\gamma_\\text{P}}{2}\\mathcal{L}_{J_{+,n}}[\\rho]\\right)\n", "\\end{eqnarray}\n", "\n", "where $J_{\\alpha,n}=\\frac{1}{2}\\sigma_{\\alpha,n}$ are SU(2) Pauli spin operators, with ${\\alpha=x,y,z}$ and $J_{\\pm,n}=\\sigma_{\\pm,n}$. The collective spin operators are $J_{\\alpha} = \\sum_{n}J_{\\alpha,n}$. The Lindblad super-operators are $\\mathcal{L}_{A} = 2A\\rho A^\\dagger - A^\\dagger A \\rho - \\rho A^\\dagger A$.\n", "\n", "The inclusion of local processes in the dynamics leads to using a Liouvillian space of dimension $4^N$. By exploiting the permutational invariance of identical particles [2-8], the Liouvillian $\\mathcal{D}_\\text{TLS}(\\rho)$ can be built with an exponential decrease in resources and the density matrix is a block-diagonal matrix in the basis of Dicke states $|j, m \\rangle$ with only $O(N^3)$ non-zero matrix elements.\n", "\n", "\n", "### Defining the Lindbladian terms\n", "\n", "The system under study is defined by creating an object of the $\\texttt{Piqs}$ class, e.g. simply named $\\texttt{system}$, whose first attribute is \n", "\n", "- $\\texttt{system.N}$, the number of TLSs of the system $N$.\n", "\n", "The rates for collective and local processes are simply defined as \n", "\n", "- $\\texttt{collective}\\_ \\texttt{emission}$ defines $\\gamma_\\text{CE}$, collective (superradiant) emission\n", "\n", "\n", "- $\\texttt{collective}\\_ \\texttt{dephasing}$ defines $\\gamma_\\text{CD}$, collective dephasing \n", "\n", "\n", "- $\\texttt{collective}\\_ \\texttt{pumping}$ defines $\\gamma_\\text{CP}$, collective pumping. \n", "\n", "\n", "- $\\texttt{emission}$ defines $\\gamma_\\text{E}$, incoherent emission (losses) \n", "\n", "\n", "- $\\texttt{dephasing}$ defines $\\gamma_\\text{D}$, local dephasing \n", "\n", "\n", "- $\\texttt{pumping}$ defines $\\gamma_\\text{P}$, incoherent pumping. \n", "\n", "The table below summarizes and compares the meaning of the keywords for an object of the Dicke class,\n", "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
 Keyword Rate $\\gamma_j$ Lindbladian $\\mathcal{L}[\\rho]$ $\\texttt{emission}$ $\\gamma_\\text{E}$ \\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{-,n}\\rho J_{+,n} - \\frac{1}{2}J_{+,n}J_{-,n}\\rho - \\frac{1}{2}\\rho J_{+,n}J_{-,n} \\right)\\end{eqnarray} $\\texttt{pumping}$ $\\gamma_\\text{P}$ \\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{+,n}\\rho J_{-,n} - \\frac{1}{2}J_{-,n}J_{+,n}\\rho - \\frac{1}{2}\\rho J_{-,n}J_{+,n} \\right)\\end{eqnarray} $\\texttt{dephasing}$ $\\gamma_\\text{D}$ \\begin{eqnarray}\\mathcal{L}[\\rho]&=&\\sum_n^N \\left(J_{z,n}\\rho J_{z,n} - \\frac{1}{2}J_{z,n}J_{z,n}\\rho - \\frac{1}{2}\\rho J_{z,n}J_{z,n} \\right)\\end{eqnarray} $\\texttt{collective}\\_\\texttt{emission}$ $\\gamma_\\text{CE}$ \\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{-}\\rho J_{+} - \\frac{1}{2}J_{+}J_{-}\\rho - \\frac{1}{2}\\rho J_{+}J_{-} \\end{eqnarray} $\\texttt{collective}\\_\\texttt{pumping}$ $\\gamma_\\text{CP}$ \\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{+}\\rho J_{-} - \\frac{1}{2}J_{-}J_{+}\\rho - \\frac{1}{2}\\rho J_{-}J_{+} \\end{eqnarray} $\\texttt{collective}\\_\\texttt{dephasing}$ $\\gamma_\\text{CD}$ \\begin{eqnarray}\\mathcal{L}[\\rho]&=& J_{z}\\rho J_{z} - \\frac{1}{2}J_{z}^2\\rho - \\frac{1}{2}\\rho J_{z}^2 \\end{eqnarray}
\n", "\n", "Note that in the table above and in $\\texttt{qutip.piqs}$ functions, the Lindbladian $\\mathcal{L}[\\rho]$ is written with a factor 1/2 with respect to $\\mathcal{L}_{A}[\\rho]$ reported in the LaTeX math equations, in order to have the Lindbladian and full Liouvillian matrix consistently defined by the rates $\\gamma_\\alpha$. \n", "\n", "We assume below that the object of the Dicke class created is named $\\texttt{system}$. Then $\\texttt{system.lindbladian()}$ creates the total TLS Lindbladian superoperator matrix that is already a quantum object (of QuTiP's $\\texttt{Qobj}$ class) with the correct dimensions and shape in the Dicke basis.\n", "\n", "### Defining the Hamiltonian \n", "\n", "Similarly, $\\texttt{system.hamiltonian}$ defines the TLS hamiltonian of the system $H_\\text{TLS}$. The total spin operators $J_\\alpha$ can be defined with the $\\texttt{jspin()}$ function. \n", "\n", "### Defining the Liouvillian (Lindbladian part + Hamiltonian)\n", "\n", "The system's Liouvillian can be built using $\\texttt{system.liouvillian()}$, which includes the Lindbladian part and the Hamiltonian, if they have been defined as attributes of the $\\texttt{system}$ object. The system's Lindbladian (the Liouvillian without the Hamiltonian) can be built using $\\texttt{system.lindbladian()}$. \n", "\n", "Some important properties of the Piqs object can be visualized by simply calling $\\texttt{system}$ (i.e. the object's name) to screen. \n", "\n", "Below we give basic examples on the use of $\\texttt{qutip.piqs}$. In the first example the incoherent emission of $N$ driven TLSs is considered. In the second example the two-level system ensemble is a subsystem coupled to another subsystem, a bosonic cavity. Similar considerations apply to the coupling to different kinds of subsystems (a single qubit, another two-level system ensemble). \n", "\n", "#### Limitations\n", "The Liouvillian derived with $\\texttt{qutip.piqs}$ is permutationally symmetric. It should not be used to derive information about the full Liouvillian spectrum, including the Liouvillian gap. " ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "import matplotlib as mpl\n", "from matplotlib import cm\n", "\n", "from qutip import *\n", "from qutip.piqs import *\n", "\n", "import matplotlib.animation as animation\n", "from IPython.display import HTML\n", "from IPython.core.display import Image, display" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## $1$. $N$ Qubits Dynamics\n", "We study a driven ensemble of $N$ TLSs emitting incoherently,\n", "\n", "\\begin{eqnarray}\n", "H_\\text{TLS}&=&\\hbar\\omega_{0} J_{z}+\\hbar\\omega_{x} J_{x}\n", "\\end{eqnarray}\n", "\n", "\\begin{eqnarray}\n", "\\dot{\\rho} &=& \\mathcal{D}_\\text{TLS}(\\rho)= -\\frac{i}{\\hbar}\\lbrack H_\\text{TLS},\\rho \\rbrack+\\sum_{n=1}^{N}\\frac{\\gamma_\\text{E}}{2}\\mathcal{L}_{J_{-,n}}[\\rho]\n", "\\end{eqnarray}" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "N = 20\n", "system = Dicke(N = N)\n", "[jx, jy, jz] = jspin(N)\n", "jp = jspin(N,\"+\")\n", "jm = jp.dag()\n", "w0 = 0.5\n", "wx = 1.0\n", "system.hamiltonian = w0 * jz + wx * jx\n", "system.emission = 0.05\n", "D_tls = system.liouvillian() " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The properties of a given object can be updated dynamically, such that local dephasing could be added to the object $\\texttt{'system'}$ symply with " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [], "source": [ "system.dephasing = 0.1\n", "D_tls = system.liouvillian() " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Calculating the TLS Steady state and steady expectation values is straightforward with QuTiP's $\\texttt{steadystate}()$ and $\\texttt{expect}()$ [9]. " ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [], "source": [ "steady_tls = steadystate(D_tls)\n", "jz_ss = expect(jz, steady_tls)\n", "jpjm_ss = expect(jp*jm, steady_tls)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Calculating the TLS time evolution can be done with QuTiP's $\\texttt{mesolve}()$" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [], "source": [ "rho0_tls = dicke(N, N/2, -N/2)\n", "t = np.linspace(0, 100, 1000)\n", "result = mesolve(D_tls, rho0_tls, t, [], e_ops = [jz])\n", "rhot_tls = result.states\n", "jzt = result.expect[0]" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "### Visualization" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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fffTRiLaVIu/zzz+PdhMirry8XObn50sAEoAsKSnxery0tFTm5+f7nZebmyvLy8ullFI2NzfL3NxcWVtb675mc3Oz+3ybzeY+z263y6KiIvfn+fn57uvX1tZKAO5zpZTSZrPJysrKgG3Pzc11f15bW+v1dfXaHaxt4XyvgRQWFgZ9XGO322Vpaamh55q5biih3ssAqqWB37Hc9sICtJtoBzvG1NzcjN7eXkyaNAnJyck45ZRT0NjYGNnG0rC59LlL/Y7duOBG3LHkDnT0duCqF6/ye/y2RbfhtkW3oaGjAYXr/bcw/1Hej7D8jOU41HII33/1+36P33v+vbh23rXY07AH87Lmhd32wsJC9xbqFRUVWLZsGWw2W9CB+KqqKtTU1KC4uBjFxcVex4uKiry2ZC8qKoLD4UBNTQ2amprgdDq9NhosKChAZWUlAGUyQH5+PtasWYOSkhJ3byZQj0LbeLCiogL5+fmGymGB2hZokkKo71WPy+VCZmZmyLa4XC44nU7DPSYAWLJkSdD2DjcGkwV0dXUBgFcpL5zp4sePHwcATJw4EUIIjB8/Hg0NDRFsKZF5hYWFWLVqFcrLy4MGk/bLVAsUXy6XC2vWrPHbFDDULDVA2YJ92bJlKCkpQWlpqbsdBQUFqK6udj+vvLwc+fn5WL16NYqLi91t8p28YbRt4X6vg1FVVQW73W4oUDU2m81SMycZTBagBVMkekwAkJGRAQDIyspijymGvXPbOwEfS05MDvp4VnJW0Mez07ODPj6Y3pIel8sV9Bc7oPRsPEPC9/zZs2fj6aefdm8cKIRwnxfqZtP8/HxkZmaiqqoKVVVV2L9/PwAEDIZVq1Zh1apVcLlcuPzyy1FWVoZVq1aZbls432sggcJj7dq1qKysdH8v69at8+stVVVVobS0FEuWLPF6rqaystK9fb3WC9Ver/z8fK/PA70OkcTJDxagV8obM2YMhBCDDib2mGg41dTUICMjA2VlZXA6nXC5XCgrK0NZWRlWr17tfl5mZqY7TLQZfFrZbNmyZe7nVVRUuH9Rulwud8/I894b7ZewNoXa5XJ5lfU0xcXFKCkpQV5eXtCQ1MILUMLAs3ym1+5gbQvnew1G776j2tpadyA6nU5UVVV5lQe1r1deXo5t27b57QjsO8tv3bp17u9Hm07u+flwYDBZgF4pTwiBcePGmSrl+QbT+PHj2WOiYaXds1ReXu6eJVdaWorKykqvcpsWJhkZGSguLnbP3Nu0aZP7eEZGBtatW+cuSxUWFrpXM/BdsWD79u2orq5GRkYGVq5cqTu+UlRUpPtLW09JSYm7DZ5jY3rtDtU2s99rMHr3HTkcDlRXV6OiogIlJSXYvn277nUcDgccDodf2bOkpMTrvqjVq1e7ZzlWVFT4fT4sjMyQ4MfQzsrbsWOHBCA3bNjgdXzixInS4XAYvs6TTz4pAchjx45JKaW844475Pjx4yPaVoq8kTgrz4qam5ul3W6PdjMGraioyGuGoRGrVq1yzwDcvn2712O+M/I8Z/MVFRX5fR5MpGblscdkAXpjTMDJPZmM0npMWpli/PjxaGpqwsDAQIRaShS71qxZE3LFhFhg9r6jmpoauFwubNu2ze8eqIqKCixfvtzrWG1trbucWVBQ4Pf5cODkBwvQG2MClJl5ZoMpOTkZSUlJAJSAklKitbUV6enpptr03nvvYcOGDfjFL35h+lwiKykrK0NxcTHy8vKGZBbccMvNzcWaNWtMPV9vzA1QxpO0SQ8a3zGo4RpX8sQekwXojTEBSo/JzBiTy+Vyjy8BcAeK2eVGpJS47bbb8Jvf/AaPP/64qXOJrKaoqAjNzc0jIpQ0BQUFEdn2wsyU8uHEYLKASJbyPGcbacHU0tJiqj21tbXu6bR/+9vfTJ1LREOvqKgoIqHi2zuyCgaTBQQq5ZkNJt97RcINpp07dwIAli5dil27dqGvr8/U+UREg8FgsoBApTyz08VbW1uRmprq/jzcYPr8888BADfccAN6enpQW1tr6nwiosFgMFlApEp57e3t7qWMgMEF06xZs3DOOecAUHbFJSIaLgwmCwhUyhs7diw6OzsNXydSwbR3717MmzcPp512GgBg9+7dps4n85RbPIhiVyTfwwwmCwhUyotWMB0+fBjZ2dlISUlBZmYm93QaYomJiab+PxNZUWdnJxITEyNyLQaTBQxVKe+UU05BYmKiqWDq6enBsWPHMG3aNADAjBkzGExDbOLEiThy5Ag6OjrYc6KYI6VER0cHjhw5gokTJ0bkmrzB1gK6uroQHx+PhATv/x1jx45Ff38/ent7Q/4l0t/fj66uLq9gEkIgPT3dVDB9/fXXAODeATc7OxsHDx40fD6Zl5aWBgCoq6tDb29vlFtDZF5iYiImTZrkfi8PFoPJArq7u/16S4ASTICxLrLWs0pJSfE6bjaYjhw5AgDuHlN2djbef/99w+dTeNLS0iL2Q00U61jKs4Curi6/8SXgZDAZKedp08o9e0xAZIKpubnZ1LR1IqLBYDBZQFdXl26PSdss0MjAeKSCSdsFd9KkSQCAKVOmAACOHTtm+Bqe13r22WfR1tZm+lwiGr0YTBZgpJQXivbLf7DB5LunkzaYqQWWGd/73vfwwx/+EPfcc4/pc4lo9GIwWYCVSnkul8trhfIJEyYAMB9MR48exdtvvw0AeOmll9DT02PqfCIavRhMFmClUp7venvh9pg2btwIKSUefPBBdHZ2+u3qSUQUyIgIJiFErhCiSAiRr/13KM+LtEiU8gIFU2pqKlpbWw3fH+MbTOH2mGpqapCamoof//jHAIB3333X1PlENHrF/HRxIYQdQImUssDjWLkQwimlDLhhSbjnDYWhLOWlpaVBSon29na/qeR6fLfOGDt2LFJTU1FfXx/yXE87d+7EwoULkZWVhdmzZ7tXLCciCmUk9JgcAHy3ZywFEGqjkXDPi7ihLOVpq423trYaaovvZoOAUs4z02OSUmLnzp0488wzAQALFizgQrBEZNhICKZCADU+x6rV40NxXsQNZSlPu2nTTDB59pgApZxnJpjq6+vhcrnci8DOnz8fe/bs4b5ORGRITAeTEMIGwA6gyfO4lNKlPq67xWO45w2VoSzlaT2mEydOGGqLXjCZ7TFpSxjNmjULgBJMvb297l1xiYiCielgApAJnAwUHYECJtzzhkSkSnkJCQnuad4aM6W8gYGBIQmm2bNnAwAOHDhg+BqaP/3pT5gwYQIef/xx0+cSUWyK9WCyhX5KZM5TZ+1VCyGqzU4ECCVQMJkt5fn2lgBzpby2tjYMDAzojjHV19djYGAg5DWAk8E0c+ZMACcDymwwdXZ24r777kNDQwOKi4tNT8AgotgU68E0bKSUZVLKPCllnjaFOlICjTHFx8cb3qsnUDCZKeW5XEoH0rfHlJWVhf7+fsPlwIMHDyItLc19nalTpyIhIcF0MP31r39FU1MTfve736Gvrw8vvviiqfOJKDaNiGBSx4yG7bxICzTGBBjfkylUMBnpMQUKpvHjxwMAGhoaQl4DUHpGWi8JABISEpCdnW06mDZt2gSbzYYf/ehHmD9/PjZu3GjqfCKKTbEeTNr9RpmeBz0CJ9D9SOGeF3FSyoA9JsD4LraRKOVp6+Tp9ZgA48F08OBBdxlPM3v2bNPB9M477+CSSy5BfHw8Lr30UnzwwQfcr4hoFIjpYFInLzjhP2aUCcAV6EbZcM8bCtoackMVTMnJyYiLizNVyvMdY9KCqbGxMeQ1AOCrr77CjBkzvI7NmjXLVDA1NzejtrYW559/PgDg4osvRnt7Oz755BPD1/DU2dnJ3WGJYkRMB5OqCkCez7Fc9fhQnBdR2rbqgUp5Y8eOHVQpTwjhXpYolEiU8rq6uuByuTB16lSv4zNnzkRdXR26u7tDXgOA+4bchQsXAgAWL14MAGEF03//939j3LhxuPrqq9Hf32/6fCIaXiMhmIoBLPM55lCPA1BKdEKIWiFEkZnzhoMWTIF6TMnJyYZ7TIGWHEpNTR305AfAWDBp+zZp+zlpsrOzAZzcuj2UXbt2AQDOOOMMAIDdbse4ceNMB9OBAwdw//33Y+bMmdi4cSOeffZZU+cT0fCL+WBSy3LFQogSIUShEGIVlDXwfMtxmWGeN6S0HsRQlfIAGO4xaWNM6enpXsfT0tKQkJBgqJSnBdPkyZO9jms9KG2H3FA+/fRTpKWluQMtLi4OCxcuNL3m3p/+9Cf09vbi3XffxZIlS/Db3/6WJT0ii4v5RVwBQEpZA//lhTwfdwHI0Dke9LzhYKSUp/VkggkWTGlpaYZLeWlpaYiPj/c6LoTA+PHjDfWYjh49CsC/x6QFU11dXchrAEownXHGGRBCuI+dddZZWLduHaSUXseDefnll3HZZZdhxowZWLFiBe644w7s2rXLvY4fEVlPzPeYYl2kSnltbW1Be0xGS3m+ZTxNVlaWqWAK1GMyGkxffPEF5s+f73XszDPPhMvlwuHDhw1d4/Dhw9i7dy+uueYaAMC3v/1tAMDf//53Q+d7eumllzB9+nRcccUVhieBEFF4GExRFiqYjJTyBgYG0NnZOehSXqhgMlPK0zYY1GRmZmLMmDGGgqmtrQ319fWw271XhtIWhd27d2/IawAn94C65JJLAABTpkzBggULUFVlbn7Lrl27cOuttyI1NRXvvPMOHA6HqfOJyBwGU5QZGWMKNStPe3ywpbzm5ma/qeIaM6W8zMxMvzX7hBCYOnWqoTEmbVq5tsaeZu7cuQDMBVN6ejrOOuss97H8/Hxs2bLF/QeBEQ899BCSk5PxwQcf4D/+4z/wyiuvhL0jb2trK+/FIgqBwRRlocaYjJTyAq0srhnuUp5vGU8zdepUQz0mbRVy32CaOnUqkpOTDQfTBx98gAsuuMBrzGzp0qXo6urCv/71L0PXaGxsxKuvvoqVK1ciMzMTP/nJT5Ceno7f/OY3hs7XSClRXFyMjIwMTJgwAX/9619NnU80mjCYoiwSpTwjwWRke3UjpbxQ1zh27JjfxAfNYIMpLi4Op556qqFg6uzsxO7du933P2nOO+88ADAcTBUVFejr68NNN90EQOl9Ll++HK+++ira2toMXQMAnnrqKaxduxbLly/H3LlzsXz5cuzYscPw+Z5cLpfhdQuJYhGDKcqMlPK6urqCruwdKpjS0tLQ398fsnzlu626p/Hjx6O/vx8tLS1BrxGqx2SklOd0OpGcnAy9xXLnzp1rKJg+++wzDAwMYNGiRV7HJ06ciFmzZuGjjz4KeQ0AKC8vx7x587yuc/PNN6OjowOvvfaaoWucOHEC//7v/478/Hy88MIL2LhxI2w2G+666y5TU9d7enrcPbfMzEzceeedLAvSiMRgijIjpTzP5+kx0mMCgq8w3tfXh9bW1oBjTEZvsj169GjAHtO0adPQ1tYWcrxr//79mD17tu6U8Llz52L//v3upZwC+fjjjwHAa3xJc+655xoKpo6ODmzZsgXXXnutV1suvPBCZGdnY/369SGvAQBPPvkkmpub8eijj7qn3v/yl7/EBx98gDfeeMPQNaSUuPXWW/HMM8/gzjvvxMqVK/H73/8eK1asMBVuUkq8+eabuOGGG3Ddddfhueee42oYZDkMpigzUsoDgu/JZKTHBARfyFULrWClPCD4enltbW1ob28P2mMCQk8Z379/v9+MPM3cuXPR398fcjfcTz75BCkpKbrXOffcc3Ho0KGQq1C8//776OnpQX5+vtfxuLg43HDDDXjrrbdChuzAwACeeuopFBQUeJUVV6xYgZkzZ+Kxxx4Ler7m+eefx8svv4xf/vKX+N3vfoc//OEPeOihh/Diiy/imWeeMXSN/v5+/PjHP8bVV1+Njz76CJ999hlWrFiBb37zm4buldN0d3fjD3/4A6666ipccskluPfee7Fv3z7D53tqbW1FXV0d+vr6wjqfRiYGU5QZKeUBgwsmI1tfBFqOSGNkvbxAyxFpjASTlNLdY9JjdGbexx9/jDPPPBNxcf5v8XPPPRcAQvaaqqqqkJSUhG984xt+j91www3o7u7G3/72t6DX2Lx5Mw4dOoQf/vCHXscTEhJw9913Y8uWLSFn+LW1tWHVqlVSlK47AAAgAElEQVS44IILcP/997uPP/DAA8jPz8dPf/pTOJ2hFyx58MEH8fvf/x733Xcf9u/fj3379uGPf/wjtmzZgiuuuCJkmRZQyqxnn3027rjjDhw4cAB9fX148skncfrpp+Phhx821Pvq6+tDWVkZcnNzkZaWhmnTpiEjIwM33HADtmzZYrgH2NDQgKeeegq33347vv/97+Oee+7B888/b+o+MyklamtrUVlZiVdeeQVvvfUWnE6n4U0xPa/jcrlw9OhRQ2OxFJzgC2heXl6erK6uDvv8S5+71P3vQ4cOwel04rEfPIZ7L7oXHb0duOrFq9yPHzt2DLt378aa5Wtw/5X3o6GjAYXrC72ud/z4cXzxP1/g8/LPkTI1Bd9/9ftejzc3N2PnUzvxTtk7mLxgMhz/638fzo2Tb8Sd37oTj7/0OF7r9h87ufO0O3Hj+Tfi35/6d7w/5n2/x5+48gl07u/EBTdfgIV3LkRmptcKUCi9phRoBE677jSc9oPT/MLr+eufR3Z6Np7+8GkUlRUhJycH06dPdz9ecWMFspKz8OQHT+KuZ+6C3W53L1cEAG/e9CaSE5Px+22/x/rP1uP999/HpEmTcOqppwIA3rntHQDAY/98DK/vfh1b3t+C7Oxs2GfbMTZxLDbepOz19NC7D2HT/k0AgO3btyM+IR6Xn385XrnxFQDA6qrV+PDwhwCADz/8EGlpabji/CvwwndeAADc8/d78PHRj93t+mL3F2je14wTL5zAKaecgqI3irC3UQnV/v5+fPjhh7An27H3t8qxmzfcjMMnvG8g7j/Yj/f/631s3boVaw+uRWPHyV+83d3d2PHKDpzfez42bdqEq/9yNTp7vf+IuWbuNTi1/lR8+9vfxuTVkzFv7jyvx0/rPw3P/uhZLD5vMRJuTfBb+eO2RbfhtkW34e/v/R3XvXAdpJQ4/fTT3f+P/8+c/4N3/7938fKbLyNtRRrmnz7fqzR97/n34tp512JPwx4sf2E5vtz3JTraO5CSkoKsrCxcZ7sOPbt78JfNf0HL+S1IS0vDzJkz3dd/5PJHcEH2BfjnoX/igU0PwOVy4euvv0Z9fT2klEj9IBVZfVmoO6UO3ecqf+jZMmyYOGEisrKy8Oz1z2Je1jy8secN/PrDX6O3txfNzc1obGqEq9mFnpd7gBMAFgBYorQ5Pj4eqampSEtLwyOLHkHBhQV4s+5NPPfxc+jr60NnZydaW1vR2tqKqZun4svPv0Traa3KNaDcHjFmzBiccsopuLHjRuTk5ODT9E/xee/nEEJgYGAAXV1d6O/qx+VHL8fBgwexLXkbmmxNkAPK7+TExESMxVhcVHcRpkyZgs+mfIbjSceRmJgIQAnDNKRheeJytLS04LXu1/C1/Bp9/X3o7+tHX18fklqTkHskF+PHj8fOWTvRPqYd8fHxiIuLQ1xcHCZjMr4lvoWOjg6U95fDJV3o7+tHf7/ykeJKwfy6+Rg7dixqTq1Bf1I/Hj3tUdxyyy0IhxBiu5TSd/FsPyNiSaJYpv1llpiQqPu49ksi2JiK9ldqoB5TQoLyv7m1tRWToV9ma2tXZpilpaUBOjuYa+vntZxoAQJs4Hv8+HEAcP/g+NJ6TMG+l/rjyhcP1INMGZeChISEoD3Irq4u5YcqwKK2cXFxSElJCTrm1tvbi7a2Nq8ND31lZWXh6NGjActQ/f39aKhvwNycuQF3KJ46dSq+3PZlwF5iT08PqrdWo7CwUOnpHfR+fMyYMbjyyivx2r2vobS0FEjzb0d9fT0euuUh5OXlYeycsX6Pn3nmmXjppZdw4003Ij0vHQvPWOjX09ywYQO+93+/h/jl8TjzzDPdPXkAyLBl4C9/+QvOf+58/L/3/x+qq6tx6qmnet1kfeTIERQ/WIxP+j/BmFPGYMGCBe7y8LUXX4v8u/Nxy/5bcMtLt+CrQ19h165dSE1NxeTJk1FTU4OWT1tQvrUcH9V/hK7OLvdrN2XKFJQ9Uoazp5yNf+z7B1ZvXI2GhgYcrz+OvXv3Yu+Xe3H989dj/sT5OJJyBDtTdqKjXbnvLyExARm2DPx0zU9x4RkXYmvrVvxl31/Q0dGB1rZWtJ5oxVdffYWbH7sZ6ADiF8dj4KwBd3AAys/WaWmn4ZZbbkHd9Dp8OvApBgYG0NPbg+6ubnR2dmLDhg1KpeECAHN9Xvxe4NjfjmHmzJnIODcD8ePj3a99b28vZIdSQfjnP/+JhrMagGyf808AWzdsVdpyTQLipsYhISEB8QnxSEhIQHJyMlpbW7F//3581fsVelK9f/b2Ht2L9/7+nnL+jQmIs8UhPj7e/ZEQn+C+4b05qxk98T147733wg4mw6SU/DD5sXjxYhkpP//5z6UQQg4MDOg+/uabb0oA8sMPPwx4jSeeeEICkA0NDbqPf/HFFxKAfPHFFwNeo6KiQgKQn3zyie7jAwMDMiEhQa5evTrgNZ5++mkJQB48eDDgc1JTU+Xdd98d8PF169YFbYeUUl544YXy4osvDvj4hg0bJAC5devWgM+56667ZHJysuzt7dV9fP369SFf97ffflsCkBUVFbqP//GPfwx5jcOHD8vExET54x//WPfxO+64Q8bHx8u9e/cGvMbAwIAsKCiQKSkp8sCBA16PuVwuefrpp8usrKyg/1+klPK5556TAGRBQYE8fvy4lFLKrq4u+fOf/1wCkOedd577eCD79u2TeXl5EoCcO3euvPXWW+W3vvUtmZCQIBMTE+UDDzwg29vbg16ju7tblpWVyTlz5kgA7g8hhFy6dKn805/+FPIaAwMDsqamRj7wwAPyyiuvlKeddppcsGCBvOaaa+SDDz4oP/roI9nf3x/0GlJK2d7eLt977z352GOPyeLiYnnffffJX/3qV7K8vFzu378/4M+tL5fLJWtqamRFRYVcv369fOONN+SOHTtkc3OzofOlVF6XQ4cOye3bt8sdO3bITz/9VDqdTtnY2BjwfexrYGBAtre3y2PHjsnDhw/LxsZG2d7ebui1iAQA1dLA79io/5KPxY9IBtPPfvYzOXbs2ICPb968WQKQmzZtCvichx9+WAKQnZ2duo8fPnxYApBPPfVUwGs888wzIUNl0qRJcuXKlSHb0dHREfA58+bNk8uWLQv4eElJiQQgW1paAj7nhz/8oZw4cWLAx3/xi19IIYRsa2sL+JyXXnpJApA1NTW6j99+++0yNTU16A98b2+vzMrKkt/97nd1H7/ooovk3LlzQ/7yWrFihRw7dqysr6/3Ov7ll1/KhIQE+aMf/Sjo+VJKeeDAAZmSkiLPO+889/fd0dEhr7jiCpmQkCA3b94c8hpSSvnss8/KpKQkmZycLJcuXSonT54sAcgVK1YE/f/qqa+vT77wwgty6dKlctq0aXLBggXypz/9qaytrTV0vmZgYEDu27dPvv322/K9996TjY2Nps4n6zEaTJz8EGVdXV0Bp4oDJ6eLh5r8EBcXF/A6RmblaVteBJouDoReL+/48eNISUnxKvP4CnUv08GDB5GRkeFus57TTjsNx48fd7fZ186dO5GTkxOwtAnAvTPuP//5T93HN2/ejIsvvthdBtWTkJCA66+/Hm+88YbfdP49e/Zgy5YtWLFiRciV0O+77z50dXXh4Ycfdh+TUuJnP/sZkpKS8J//+Z9BzweUjRj//Oc/41//+hcuuugi/PrXv8Y3vvEN/OMf/0BpaSkuvfTSkNcAlNmCO3bswPe+9z20t7fjoosuwltvvYVnn3026P9XT/Hx8bjpppuwadMmHD58GJ9++ikef/zxgDMtAxFCICcnB5dddhkuuugiv3FLGrnCDiYhxHeEEP83ko0Zjbq6ugKOpwDGZ+WlpKQE/AWo/YIONqbicrncYy+BhFqWqL6+3m/xVl+hgunAgQNBx3WAk4u57tmzR/dxI9tazJw5E1OmTMGHH37o99iRI0ewZ88eLF26NOg1AGV2XltbGyorK72O//GPf0R8fDxuu+22kNeYP38+HA4Hfve737lXpPjzn/+Mv/71r3jwwQcDTr/39Z3vfAevvvoqGhoacN999+H48ePYsGEDfvCDHxg637M9Tz/9NLZu3Yr169fjm9/8pqnziQbLVDAJIRYJIZ4SQmwDYAeQIYSoFkL8QQixKNT55K+7u9tQMAVbyDXYXkyAMtgfaoVxbTmiYH/dh1rI9fjx4yGDKTs7G3V1dQGn4x48eBAzZ84Meg0tmHbv3u33WEdHB/bt2+fekj0QIQTOP/983WDavHkzAOCyyy4Leg1AWXsvIyMDFRUV7mNdXV3485//jGuvvdZwqDz88MPIzs7GFVdcgZtvvhkrV67EpZdeinvvvdfQ+ZrrrrsOBw4cwPHjx3Hw4EH3Vh9EsSRkMAkh0oQQPxNCVANYDaBcSrlESvmYlPJXUpn6VwbgdiHENiHEfUKIwHUY8hKpUl6wYAJCb30RbJ08jZFSXqhgmj59Onp7e90z+DxJKXHgwIGQwTRr1iwkJSXpBtPnn3+OgYEBQxsBnn/++XA6ne49pDSbN29GRkaG7qoRvhITE3H99defnHkFoKysDMePH8ddd90V8nxNZmYmNm3ahNzcXPzv//4vvvvd7+K1117zm7ptRFxcHCZMmKB7DxdRLAj4zhVC3CCE+AeATQCapZR5UsrlUspNvs+VUu6QUt4upVwCoAXA20KIt4QQ3xm6po8MkSrlGQmmYKW8YFteaLRSnjKG6a++vl53fTtP2r1Hhw4d8nusqakJ7e3tIUt5CQkJmDNnjm4w7dq1CwBC9pgA4OKLLwYAvP322+5jAwMDeOutt7B06VLDv9jvvfdetLe344EHHkBdXR1++ctf4rLLLjNUCvRkt9uxadMmuFwu/M///E/QcTaikUz3J08I8RSAPAAOtXdkbM0TAFLKp9Ve1O0AzhFCrItMU0emSJTygu1eqwm1J5ORHlOwhVwHBgYMjTFpN83q7UKr7cMUqscEKOU8vWDauXMnkpOTDQ205+XlYcKECV6rN1RXV+PIkSP4t3/7t5Dna+bPn4/77rsPTz/9NGbMmIH29nb89re/NXw+EXnTnXIkpbx9sBeWUu4HcH/IJ45yoUp5Y8aMgRAiZI8p1F/XRkp5U6ZMCXoNz/XyfEPM5XKhr6/P0BgToN9jOnhQuXs0VI8JUILp9ddfR29vr9cNvbt27cKCBQsMlcDi4uJw1VVX4fXXX0dfXx8SEhKwYcMGJCQkuLdjN2rNmjWYPn06du7ciTvvvBNnnHGGqfOJ6KSIFKGFELMicZ3RKFQpTwgRck+mSJTyjI4xAfrr5WljRqFKeVlZWRgzZoxuj0kLJiM9ptNPPx19fX1ea+ZJKbFjxw5DY0Oab3/722hubsbGjRvR29uL559/Hvn5+SHLmr7i4+Pxk5/8BM888wzOPvtsU+cSkbdILUnULIRIk1KeEEKkA1gGYL2UkruZhRCqlAeE3l7dSDCFKuUZGWMKtpBrfb2ylFCoHpMQAtOnT9ftMR04cACpqamGQiE3NxeAUnpbsEBZoGzfvn1oampyL9JqxNVXX41p06bhkUcewf79+1FXV4eysjLD5xNR5EVq2k4GgCVqOLUAKFc/KIRQpTwg9Pbqg52V192trOlltMekNzNP6zGFCiZAGWcK1GOaOXNmyBtSAWDevHlITU31WplbWy3cTDAlJibi0UcfxdatW3H33XejoKAAV111VegTiWjIRKrHVCmlPFUIke4xVXx8hK49ooUq5QGht1c32mMKVMrTJjMMRykPUMaZtmzZ4nfcyM21mvj4eCxevNhri/StW7ciJSUF8+fPN3QNzU033QQhBPbt24e7777bUDAS0dCJVDCVCSEWSSk/Vkt5iwFcHqFrj2iDDSYpJTo6Ogz1mHp7e9Hd3e3XQwu1F5MmLS0NCQkJQUt5WngFM336dBw5cgQDAwPuKdnaPUx6ex8Fcs455+CJJ55wv4ZbtmzBOeecY/reHyEEbrrpJlPnENHQCXYf0yyjF5FS/grAfvXfLeq/udGTAYMdY+rs7ISU0lAwAfrLEhlZJw+Ae1vwQKW8jIwMJCUlBb0GAMyYMQN9fX1eO8jW19ejpaXFvRGgERdffDF6enrw7rvv4tChQ9i5cyeuvPJKw+cTkTUFG2MqMXMhNZC0f+8HwHqIAYMdYwq1e60m2EKuRntMQOD18o4fP26ojAcAOTk5AOC1Hbc2u85MMC1duhRjx47Fq6++6l4S6OqrrzZ8PhFZU7BS3jIhhASwDsAmszPsPIOK9EkpDZfytPDwZTSYgm2vbiaYAq2XZ+TmWo0WPl9++SUuueQSAOEF09ixY3HjjTfi2WefRUpKCi688ELT40tEZD2hZuV9E8ArUKaDf6ku1no918KLjL6+PkgpB1XKMxtMeqU8sz2mQKU8o8GUnZ2NpKQkr3uQ9uzZg6SkJEP3MHl68MEHkZKSgvb2djzyyCOmziUiawoWTBVSykwAcwD8CMAOAMuhE1TD0M4RSdvDJ9qlPKNjTEBkSnnx8fGYM2eOVzDt3bsXc+bMMT1xYfbs2aitrcVXX33lXvuOiGJbsGBqAgAppVNKWSalvFENqhz4BJUQolEIEXqPAPKiBdNgZuVFqpSXlJQUsh0A3JMfPBdy7evrQ2Njo+EeE6CU7L788kv357t37zZVxvOUkZGBSZMmhXUuEVlPwGAKtF6elHK/TlA9A6BKCGGuDjPKdXd3AzAWTIMt5Wk9pkClvFB7MWmysrLQ19fndZ1jx45BSolp06aFPF9z6qmnYt++fejr60NbWxv27NmDRYu4pRcRRWDlBzWoigEsAbB28E0aPSJZygu28ywQusdkdG04vZts6+rqACi70xp19tlno6enB59++ik++eQTSCndywwR0egWqRtsIaWsUTcTJIPMlPL6+/v9VtIGjPeYtOAKNMZkZOID4L1enjbtW9sq3UwwnXfeeQCU1Rq03uDixYsNn09EI9egg0ndDNAppfwYvKnWFDPBBCh7MqWnp3s9ZjSY4uPjMW7cuIClPLM9Js+ZeeH0mGbNmoVJkyZh8+bNaGxsxOmnn27qfCIauSKxiOtaAJvUgOK9SyYYHWMKtr260WACAi/kamTLC02gUl58fLypyQ9CCBQWFmL9+vXYtGkTrr32WsPnEtHIFolgWgxlQ0CnlPLpCFxv1DA6xhRse/X29nYIIQzNqAsUTEa2vNDobX1RV1eHyZMnm57q7XA4kJiYiOTkZBQVFZk6l4hGrkGX8tQVHhhIYQinlOdL21bdyIw6vRXGpZSmekzp6emIj4/3K+WFU4ZbuHAhdu3ahTFjxhheVZyIRr6ITX4g8yJVyjNSxgP0e0ydnZ3o7e01HExCCL+bbOvq6mC32w2d72vevHlhnUdEI5duKU8I8TMhxNLBXlwIsUgIcd9grzNSRaqUZzSY9HpMTU1NAIDMzExD1wCAyZMnuyc8AMChQ4dM3cNERBRMoDGmCgA3qssOrTGzBYYQIk0NtmoADgCbBt/MkclsKW8oekzhBNOsWbOwf/9+AMr4lMvlck8dJyIaLN1Snrptxe0AIIS4AcpGgBJAqZRyg9456vMcUKaMl0sp84amySOHmZUfAP0xpmgE0+zZs1FZWQkpJWprawEg7FIeEZGvkGNMUspXoKyHNxuAQwixFkAlgFIALigBdjmAKgAONdTIADMrPwDWKeXNnj0bHR0daGhocAcTe0xEFCmGJz+ogXM/gPuFEJcDeABK76hMSnn/ELVvRItUKc/oqt6pqano7u72WkFCCyaj08UBuGfQOZ1OdzDNnj3b8PlERMGENStPSrkJHDsaNK2UZ3TyQyRKeYCyLJHWQ9K2vDDTY9I249u1axc++eQTzJo1K+RafURERkXiBlsKk7ateqh7kCJZygO8VxhvampCYmKi4WsAStkuPT0d27dvR3V1Nde4I6KIYjBFkRZMoURyVh7gvZBrU1MTMjMzDd2gqxFC4Oyzz8Y//vEPOJ1OBhMRRRSDKYq6uroMLSUUHx+PxMREv1KelDJiwWTWZZddBqfTCQDIz883fT4RUSAMpijq7u42FEyA/p5M3d3dGBgYMBxM2srkLpfLfSzcYFq5ciVOPfVUFBYWIi+PdwYQUeRwSaIoMlrKA/S3Vzezsjigv2VFU1MTsrOzDZ3vacqUKdi7d6/p84iIQmGPKYqMlvIA/e3Vww0mz3Xuwu0xERENFQZTFA22lKeNFWljR6FoK4N7BlNjY6Ope5iIiIYagymKBlvKa2trA2A8mHxXBu/o6EB7ezsmTZpkotVEREOLwRRFZkt5gYLJzM2tnsF07NgxAGAwEZGlxPTkByFELoA8AE4Adii76FaFOMcGoAjKCupNADKhLD5bGercSOvu7nbvCBtKcnKye5UGzWCD6ejRowAYTERkLTHbYxJC2AGUSCnLpJRVUsoyKIvMhlrmOhNACYBaAM0AtgPYNtyhBAy+lGd2jAnQ7zFNnjzZ8PlEREMtZoMJSi+n1OdYKZTQCaUAQAaAHCllhpSyItKNMyIapbyJEye6e0rsMRGRFcVyMBUCqPE5Vq0eD0lK6ZJSOiPeKhMGO108nGCaPn06mpqa0NHR4e4xTZw40fD5RERDLSaDSR0nskMZI3KTUrrUxyO+a50QokgIUS2EqK6vr4/INSM1XdzMAqzazbRHjhxBXV0dsrKy3FtgEBFZQaxOfsgETgaRDjuUCRGB2IUQWs8qE0BTqHKeOoZVBgB5eXnSXHP1RWK6+Lhx4xAXZ/zvi+nTpwMADh06BKfTyZ1nichyYjWYbIM4twkAPINICFEuhMBwjzWZLeV1dXVhYGDAHURtbW2m90HSgunw4cNwOp0499xzzTWaiGiIxWQpbzDUsaUyn8NGJ01ETF9fH/r7+02V8oCTu94CSikv3GCqra3FwYMH2WMiIsuJeo/J5HhQk2f5TghhC1LOM8MJpbwXqeuFZHT3Wo3nnkxaSLW1tZmaKq5dZ86cOXj99dfR39/PYCIiy4lqMGn3Ipk4ZRuAtTg5fpQJwCuo1H8GHF8SQqySUq71OaxNorDDf6bfkNB6PmZKeYCyjJB2U244pTwAyMvLw8svvwwAWLhwoenziYiGUlSDSZ2uvSyM81xCCCf8x5oyAQScBq4FoRCiwuc52vLawzZ9XOsxmS3leU6AaGtrM7xyhKclS5a4g+mss84yfT4R0VCK5TGmKijLEXnKVY/rUsPIoRNc+QBqhquMB5zsMYVTytOEM8YEADfddBMWLlyIX//610hKSjJ9PhHRUIr6GNMgFAMohzqFW+VQPwC4S3vboS5dpB5uEkLYtXBSn+MAsHJYWq0yG0xaj0nbgwkIb4wJUFZ62Llzp+nziIiGQ8wGk1rOKxZClEAZe9LWzvPtDWX6nFchhChU72MaD6UcuGy4V4HQgknrCYWSlpYG4ORNtUD4Y0xERFYWs8EEAFLKGgSZrKCW5vx2wYvW2nietJKc0WBKT08HALS0tAAApJRhl/KIiKwslseYYprZWXlaj+nEiRPu8/v7+8Mq5RERWRmDKUrCLeVpwaT1nGy2wSyCQURkPQymKNFKeUZ7TFrJTgsml0uZQMhgIqKRhsEUJWZ7TPHx8UhNTWUwEdGIx2CKErM9JkAp52klPAYTEY1UDKYoMTv5AVCCiT0mIhrpGExRYna6OMBgIqLRgcEUJeH0mNLT0xlMRDTiMZiipLOzE4mJiYiPjzd8Tnp6OpqbmwEowZSUlGQq2IiIYgGDKUrM7F6rycrKQmNjIwDlPiabzQYhxFA0j4goahhMUdLV1WVqfAlQgqmpqQn9/f1oaGhARobfaktERDGPwRQlnZ2dYfWYBgYG0NzcjGPHjmHSpElD1DoiouhhMEVJOKW8CRMmAAAaGhoYTEQ0YjGYoqSzszOsUh7AYCKikY3BFCXhTn4AgMOHD6OlpYXBREQjEoMpSsLpMWmlvM8++wwAGExENCIxmKIknB7TxIkTIYRATY2yNyKDiYhGIgZTlITTY0pKSsLUqVPx7rvvAgBmzJgxFE0jIooqBlOUhNNjAoDZs2ejvb0dADBnzpxIN4uIKOoYTFESzg22AHDWWWcBAKZPn+7ePJCIaCRhMEVJODfYAsAVV1wBALj00ksj3CIiImtIiHYDRqtwS3nXXHMN3nnnHZxzzjlD0CoiouhjMEWBlDKsyQ8AIITAJZdcMgStIiKyBpbyoqCvrw8DAwPcsoKISAeDKQrC2b2WiGi0YDBFQTi71xIRjRYMpijQ7kMaN25clFtCRGQ9DKYoYDAREQXGYIoCBhMRUWAMpijQgik5OTnKLSEish4GUxR0dHQAYI+JiEgPgykKWMojIgqMwRQFDCYiosAYTFHAYCIiCozBFAUcYyIiCozBFAXt7e0QQnDlByIiHQymKGhvb0dycjKEENFuChGR5TCYoqC9vZ1lPCKiABhMUcBgIiIKjMEUBR0dHQwmIqIAGExRcOLECaSmpka7GURElsRgigKXywWbzRbtZhARWRKDKQpaWlqQnp4e7WYQEVkSgykK2GMiIgqMwRQF7DEREQXGYBpmXV1d6OnpYTAREQXAYBpmLpcLAFjKIyIKgME0zFpaWgCAPSYiogAYTMNM6zExmIiI9DGYhpnWY2Ipj4hIH4NpmDU0NAAAMjMzo9wSIiJrYjANs6NHjwIApkyZEuWWEBFZU0K0GxAJQohCAC4pZZXB5+cCyAPgBGAH4DR67mAdPXoUY8aM4RgTEVEAMR9MQoh8AE8DWGbw+XYAJVLKAo9j5UIIp5TSOUTNdDt06BCmTZvGTQKJiAKI2VKeEMIuhCiF0uNpMnGqA0Cpz7FSACWRalswe/bswdy5c4fjSxERxaSYDSYppVNK6ZBSlpk8tRBAjc+xavX4kDp48CB27tyJxYsXD/WXIiKKWTEbTOEQQtig08OSUrrUx+1Bzi0SQlQLIarr6+vD+vrt7QhGQtUAAAcCSURBVO0oKCiAw+EI63wiotFgVAUTgEzgZBDpCBhMUsoyKWWelDJvwoQJYX3x+fPnY+PGjcjOzg7rfCKi0WC0BRPvaiUisrjRFkxERGRxUZ8uHmxcR0dTkDKcma9pi8R1iIgo8qIaTNo9RSZO2QZg7SC+pHafUiYAdzCpkyI8HycioiiJajCpN7QaujE2Ql/PJYRwwn+sKRPKyhEMJiKiKBuNY0xVUJYj8pSrHicioigbKcGUCZ0Zd0IImxCiVghR5HG4GP69NId6nIiIoizqkx/CpY4LrYZy75ENQIkQogBApZSywuOpXvtLqOW8YiFECZQxK23tPJbxiIgsQEgpo92GmCOEqAdwcBCXyALQEKHmjAZ8vczh62UOXy9zBvN6zZRShlyhgMEUBUKIaiml7zgXBcDXyxy+Xubw9TJnOF6vkTLGREREIwSDiYiILIXBFB1mt+oY7fh6mcPXyxy+XuYM+evFMSYiIrIU9piIiMhSGExERGQpDCYiIrKUmF35IdYIIXKhrNHnhLLahFNKyfX5AAghCqG8Jjnqf0s9V+9QV/koAlABoAnKah4OKKt8jLrX0OjrwfecQl3lpRJAdaDtbkb7e0z9GXTpfa9G3kcRf69JKfkxxB/q/6hKn2PlAOzRblu0PwAUAsj1+NwGoBZAkc/rJz0+mgEURrvtUXzNQr4efM95fd+VPq+X50et0dd0pH4AyFe/3/wA77Wg76OheK+xlDc8HABKfY6VwtxeVCOVXUpZo30ilb9oS+D/ehUAyACQI6XMkN7rIY5GoV4PvudOckJ5vRZD6ZVrHw54L+g8qt5jQgi7EKIUSrA0BXiakfdRxN9rDKbhUQigxudYtXp81FLLJ8s9NmrUVKmPe+1uLKXknlkeQrwefM+dtF1KWSWlrJFSOrUPAPD8o0j9fNS8x9TXwSGlDHZfkpH3UcTfawymIab+0vX7i0TtGZjdWn5EUV8Du/pBEcL3nDe9X7xCiKIQv5BHPSPvo6F6r3Hyw9DLBE7+j9Jhxyje0l1KmaFzOB/+Owrb1QFaQHlNm0Z6qSWEYK8H33NBCCHyofxF74vvMW9G3kcw8BzT7zUG09Dz28CQQnIAWOPxeRMASO+ZeuVCCIzSXxyhXg++54LLlVKu9TnG95g/I++jIXmvsZRHlqLuNtzk+YtDrfv7ll1G60A+X49BUHtEfn/B8zW1FgbTMNEZ4Ccfaj3aIaUsMPB0J5TSC19Xhd/rwddG12qok2sM4HsMxt5HkX6NGExDT/vrzGuLd4//kaO21q+jBMDlvgeFEKt0nqsNto6qgXzA0OvB95wO9fvP1RsP4XtMl5H30ZC81xhMQ0z9IXDCvxabCf8B/lFLvZ+i2PeXhtqLKtGZ3aP9IIyq18/I68H3XED5APRCie8xHUbeR0P1XmMwDY8qKMt1eMqF8ZLCiKaOK5V4vomFEPlCCLt6zKHzBs8HUBNkNtCIZOL14HvO3xLojy/xPRaYkfdRxN9rDKbhUQzvO8wBZeZZcRTaYike03NtQohc9SMfwDKPXxRNnn/NqmUCB4CVw9xcqzDyevA95y/YCgej/T2WCf0ZdkbeRxF/r3GjwGGiLnK4HMA2KD8gNXIULA4ZjPrD3xzgYaeUMsfjudpCr+Oh/ACVjOKSlKHXg+85b2q5GFJKR4DHR9V7TP35Ww3le9ZmK1ZBWffOc9p8yPdRpN9rDCYiIrIUlvKIiMhSGExERGQpDCYiIrIUBhMREVkKg4mIiCyFwURERJbCYCIahdQbmWuj3Q4iPQwmotFpOXTWjSOyAgYT0eiUj9G9bh5ZGIOJaHTKBVAZ7UYQ6WEwEVmYEMImhAi6i6oQolRbBy7E8/LV52qBtEz9PDcijSWKkIRoN4CIgiqCssV3MC4opbmg1EU1q9RN8TIDLWZKFG3sMRFZnIEVrtfA3GZ2BeD4ElkYg4nIotRtGCp8juWrGyu6qRvZbTNx6XxwfIksjMFEZF2ZOr0lv96OurGioR6Qx3hS9eCbRzQ0GExE1pWjc8yuE1bLpJQ1Bq/ptV24ulkckaUwmIisK9czONRJC4U+W4AXAdju8bldLfeVCCEK1X+Xe1xzCbx7S15lQSIr4Kw8IutyAihXg6UAynbfDvVYKZQeVb6UcrHHOXYpZZUaaMullMs8g0y1HTBXAiQaTtxancii1HDZBMAOJUBWSild6n1IWqg49Gbtqfc+bZNS+k6eyAWwGsA6AC51CjmRpTCYiEYgIcR2AJdrY0lEsYRjTEQjhDq+lKuW7uweExwKo9w0IlPYYyIaIdQAsgOoUf/bBAC+5Twiq2MwERGRpbCUR0RElsJgIiIiS2EwERGRpTCYiIjIUhhMRERkKQwmIiKyFAYTERFZCoOJiIgshcFERESW8v8DniYHIS9UOYYAAAAASUVORK5CYII=\n", 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