{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Biomechanical analysis of vertical jumps\n", "\n", "> Marcos Duarte \n", "> [Laboratory of Biomechanics and Motor Control](http://pesquisa.ufabc.edu.br/bmclab) \n", "> Federal University of ABC, Brazil" ] }, { "cell_type": "markdown", "metadata": { "toc": true }, "source": [ "

# Contents

\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Python setup" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "ExecuteTime": { "end_time": "2021-02-09T21:13:02.552800Z", "start_time": "2021-02-09T21:13:02.348546Z" } }, "outputs": [], "source": [ "import numpy as np\n", "from scipy.integrate import cumtrapz\n", "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "from IPython.display import YouTubeVideo" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Introduction\n", "\n", "A vertical jump is the act of raising (in the vertical direction) the body center of gravity using your own muscle forces and jumping into the air ([Wikipedia](https://en.wikipedia.org/wiki/Vertical_jump)).\n", "\n", "Besides being an important element in several sports, the vertical jump can be used to train and evaluate some physical capacities (e.g., strength and power) of a person.\n", "\n", "In this notebook we will study some aspects of the biomechanics of vertical jumps." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Center of gravity\n", "\n", "Center of gravity or center of mass is a measure of the average location in space of the body considering all body segments, their mass and their position. \n", "From Mechanics, the exact definitions of these quantities are: \n", "\n", "- **Center of mass (CM)**: The center of mass (or barycenter) is the unique point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. SI unit: $m$ (vector). \n", "- **Center of gravity (CG)**: Center of gravity is the point in an object around which the resultant torque due to gravity forces vanishes. Near the surface of the earth, where the gravity acts downward as a parallel force field, the center of gravity and the center of mass are the same. SI unit: $m$ (vector).\n", "\n", "The mathematical definition for the center of mass or center of gravity of a system with N objects (or particles), each with mass $m_i$ and position $r_i$ is:\n", "\n", "\$$\n", "\\begin{array}{l l} \n", "\\vec{r}_{cm} = \\dfrac{1}{M}\\sum_{i=1}^N m_{i}\\vec{r}_i \\quad\\quad \\text{where} \\quad M = \\sum_{i=1}^N m_{i} \\\\\n", "\\vec{r}_{cg} = \\dfrac{1}{Mg}\\sum_{i=1}^N m_{i}g_{i}\\vec{r}_i \\quad \\text{where} \\quad Mg = \\sum_{i=1}^N m_{i}g_{i}\n", "\\end{array}\n", "\\label{}\n", "\$$\n", "\n", "If we consider $g$ constant:\n", "\n", "\$$\n", "\\begin{array}{l l} \n", "\\vec{r}_{cg} = \\dfrac{g}{Mg}\\sum_{i=1}^N m_{i} \\: \\vec{r}_i = \\dfrac{1}{M}\\sum_{i=1}^N m_{i}\\:\\vec{r}_i \\\\\n", "\\\\\n", "\\vec{r}_{cg} = \\vec{r}_{cm}\n", "\\end{array}\n", "\\label{}\n", "\$$\n", "\n", "This means that how much a person jumps is measured by the displacement of his or her center of gravity, and not by how much the feet was raised in space. For instance, the following Youtube video entitled \"Highest Vertical jump 62 inches\" (157.5 cm!) shows a great vertical jump, but in fact its height is 'just' about half of that:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "ExecuteTime": { "end_time": "2021-02-09T21:13:02.651415Z", "start_time": "2021-02-09T21:13:02.553778Z" } }, "outputs": [ { "data": { "image/jpeg": 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\n", "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "YouTubeVideo('NUyql5IFTNY')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "So, the true measurement of the height of a vertical jump is the displacement of the center of gravity in the vertical direction. A problem is that to measure the center of gravity position is usually too complicated because we would have to estimate the position of each body segment. \n", "Instead, there are alternative methods with varying degrees of accuracy for measuring the height of vertical jump ([see here for a list](http://www.topendsports.com/testing/equipment-verticaljump.htm)): \n", "\n", " - [**Sargent Jump Test**](http://www.brianmac.co.uk/sgtjump.htm). How high you can reach an object. \n", " - **Flight time measurement**. From Newton's laws of motion, the height of a jump is related to the flight time under controlled conditions. One can measure the flight time using a [contact mat](http://www.probotics.org/JustJump/verticalJump.htm), light sensor, [accelerometer](http://www.jssm.org/vol11/n1/17/v11n1-17pdf.pdf), video (the number of frames the jumper is in the air), attaching a cable to the jumper's waist and measuring the displacement of this cable, etc. \n", " - **Force platform measurements**. A device called [force platform](https://en.wikipedia.org/wiki/Force_platform) which measures the forces applied on the ground as a function of time can also be used to measure the flight time and then the height of vertical jump. But the real utility of the force platform is that it can actually measure the force produced by the jumper to find the displacement of the center of gravity using Newton's laws of motion. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Measurement of the jump height from flight time\n", "\n", "We can estimate the height of a jump measuring for how long the jumper stayed in the air during the jump (flight time). At the vertical direction and ignoring the air resistance, the only external force that acts on the body in the air is the (constant) gravitational force, with magnitude $P = -mg$, where $g$ is the acceleration of gravity (about $9.8 m/s^2$ on the surface of the Earth). Using the equation of motion for a body with constant acceleration, the height of the center of gravity of a body in the air at a certain time is: \n", "\n", "\$$\n", "h(t) = h_0 + v_0 t - \\frac{gt^2}{2}\n", "\\label{}\n", "\$$\n", "\n", "At the maximum height ($h$, the jump height), the vertical velocity of the body is zero. We can take use this property to calculate the jump height from the time of falling:\n", "\n", "\$$\n", "h = \\frac{gt_{fall}^2}{2}\n", "\\label{}\n", "\$$\n", "\n", "Because the time of falling is equal to the time of rising, $t_{flight} = t_{rise} + t_{fall} = 2 t_{fall}$, the jump height as a function of the flight time is:\n", "\n", "\$$\n", "h = \\frac{gt_{\\text{flight}}^2}{8}\n", "\\label{}\n", "\$$\n", "\n", "This simple equation is the principle of measurement of the jump height any an instrument that measures the time the feet is not in contact with ground (using a pressure mat or a photocell sensor). \n", "There are (were) even shoes using this principle of measurement:" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "ExecuteTime": { "end_time": "2021-02-09T21:13:02.740049Z", "start_time": "2021-02-09T21:13:02.652681Z" } }, "outputs": [ { "data": { "image/jpeg": 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\n", "text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "YouTubeVideo('QXnQ4ENjTuY')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Beware: the flight time you measure not always is the flight time you want\n", "\n", "However, the flight time, measured as the time without contact with the ground, during a jump is not necessarily equal to the flight time of the body center of gravity (which is the measure we need to estimate the actual height jump). \n", "\n", "For example, if the jumper flexes knees and hips at the landing phase, the measured flight time will be larger but not the flight time of the body center of gravity. Because that, a more accurate method is to use a force platform as we will see now." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Measurement of the jump height using a force platform\n", "\n", "Let's draw the [free body diagram](https://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/FreeBodyDiagram.ipynb) for a person performing a jump:\n", "\n", "
\n", "\n", "So, according to the Euler's version of the Newton's second law (for the motion of the body center of mass), the dynamics (as a function of time) for the body center of mass during a jump is given by:\n", "\n", "\$$\n", "GRF(t) - mg = ma(t)\n", "\\label{}\n", "\$$\n", "\n", "Where $GRF(t)$ is the ground reaction force applied by the ground on the jumper, $m$ is the subject mass, and $a(t)$ the center of mass acceleration." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Force platform \n", "\n", "Force platform or force plate is an instrument for measuring the forces generated by a body acting on the platform. A force plate is an electromechanical transducer that measures force typically by measuring the deformation (strain) on its sensors and converting that to electrical signals. These electrical signals are converted back to force and moment of force using calibration factors. \n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Usually the force platform is placed on the floor and we are interested not in the force applied on the force platform, but in its reaction, the force that the force platform applied on the jumper, which, according to Newton's third law of motion, has the same magnitude and line of action but opposite direction. Because of that, usually the forces measured by the force platform are referred as ground reaction forces. \n", "\n", "Most of the commercial force platforms are able to measure the vectors force, $[F_X,\\, F_Y,\\, F_Z]$, and moment of force, $[M_X,\\, M_Y,\\, M_Z]$, from which the [center of pressure](https://en.wikipedia.org/wiki/Center_of_pressure_(terrestrial_locomotion) (COP, the point of application of the resultant force on the force plate) can be calculated $[COP_X,\\, COP_Y]$. Because of that, these force platforms are known as six-components. Force platforms that can measure only the vertical force component and two moments of force (or the two COPs) are known as three-components.\n", "\n", "Read more about force platforms in chapter 5 or Winter's book and in Cross (1999). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Ground reaction force during vertical jump\n", "\n", "Here is a video from Youtube of a person jumping and a plot of the ground reaction force measured with a force platform:" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "ExecuteTime": { "end_time": "2021-02-09T21:13:02.824992Z", "start_time": "2021-02-09T21:13:02.741306Z" } }, "outputs": [ { "data": { "image/jpeg": 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"text/html": [ "\n", " \n", " " ], "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "YouTubeVideo('qN3apht8zRs')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here is a plot of the vertical component of the ground reaction force measured with a force platform during a vertical jump with countermovement:" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "ExecuteTime": { "end_time": "2021-02-09T21:13:02.946896Z", "start_time": "2021-02-09T21:13:02.825938Z" } }, "outputs": [ { "data": { "image/png": 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duiWOV6/OT0wi5STdKhZnAP8CDnf3ddHax9cB9wDzga+6+zt5iFFERERiamtDkgzQpg20bZt8f8+eieMVK/IXl0i5SFdisRNwrbuvg883Arksesx5So5FREQKI93oMSQnyMuX5z4ekXKTLkHuBHxUr63u/N3chCMiIiJNSbfEG2gEWaS1mlrFwhtpr812ICIiIpKZpkaQe/VKHCtBFmm+praH/ruZbUzR/piZxTa5xN19+yzGJSIiIo3QCLJIbqVLkO/MWxQiIiKSsaZGkHv0SByvWAFbtoTJfCKSmXRbTX8v2y9mZtsCdwH9gS3ALe5+rZn1BP4KDAEWAEe7+8roMecAJxPKOs5y9yej9nHAHYRa6ceAH7h7YyUhIiIiZaOpEeT27cNayDU1ITleswa6d89beCIlL99/T24GfuzuOwMTgNPNbBTwC+Apdx8BPBWdE913LLALMAm4wczqFrO5EZgMjIhuk/LZERERkUJpagQZVIcs0hppa5DNrA+wyd1XxdpOqnfZWnf/f5m8mLsvAZZEx2vN7E1gEHA4UBVddidQDfw8ar/X3TcA881sHrCnmS0Aurr7C1FMdwFHAI9nEoeIiEgpS7fNdJ2ePeH998PxihWwww65j0ukXKTbSW9/wkYh3wXujdraAn8grG5h0aVuZkvc/bnmvLCZDQHGAi8C/aLkGXdfYmZ9o8sGAdNjD1sYtW2Kjuu3p3qdyYSRZvr160d1dXVzwmy1mpqavL9moaiv5Ul9LT+V0k8o377OmtWN8CsU1q9fRXX1rAZ9NdsdCMXI06bNpqZmZf4DzZFy/b6mUkl9LSbpRpBPAp5393tT3Pd1YA4hSf4NoUY44wTZzLoA9wM/dPc1ZtbopSnaPE17w0b3W4BbAMaPH+9VVVWZhpkV1dXV5Ps1C0V9LU/qa/mplH5C+fY1XoPct293qqqqGvR12DB45ZVwPHjw7pTTP0O5fl9TqaS+FpN0NchfAu5u5L4l7v6+uy8gJLr7ZPqCZrZV9Jg/u/sDUfNHZjYgun8AsCxqXwhsG3v4YGBx1D44RbuIiEjZa2qSHqgGWaQ10iXIg4C367VtAZ4EVsfaltJIeUN9FoaK/wi86e5Xx+56GDgxOj4ReCjWfqyZdTCzoYTJeDOicoy1ZjYhes4TYo8REREpa5lM0tNayCItl67EYjOwVbwhWkbtkHrXtSckzpn4EnA88JqZzYrazgUuB6aY2cnAB8C3otd73cymAG9E8Zzu7nW7+J1KYpm3x9EEPRERqRCZjCArQRZpuXQJ8nxgPDC1iefYA3gvkxeLJvI1VnB8YCOPuRS4NEX7TGB0Jq8rIiJSTpo7grx8eW7jESk36Uos/g6cEW3ikZKZ9QZOj64VERGRPFANskhupUuQryaM9v7bzL5uZp//jRrVBB9OWLnComtFREQkD1SDLJJb6baaXm5mBwMPAA8CtWb2cXR3H6AtobRikrvrwxsREZE8UQ2ySG6l3UnP3edE2z0fBRxAWE7NgA8Jm4jc7+6bch6liIiIfE41yCK5lTZBBnD3zYSd9FJtGCIiIiJ5lulW03VWrIDaWmjbNrdxiZSLdDXIIiIiUoTiJRaNjSC3bw+9e4fjLVtg2bLU14lIQ0qQRURESkwmJRYAg2LbeC1cmLt4RMqNEmQREZESk8kkPUhOkBctyl08IuVGCbKIiEiJ0QiySG4pQRYRESkxmY4gDx6cONYIskjmlCCLiIiUmJaMICtBFslco8u8mdm/mvE87u4HZiEeERERaUJLapBVYiGSuXTrILcBPMPnsSzEIiIiIhnIdARZJRYiLZNuq+mqPMYhIiIiGWrpKhbuYBrSEmmSapBFRERKTKYjyN27Q6dO4XjdOli9OqdhiZSNJreajjOzHsAIoGP9+9z9mWwFJSIiIo3LZKtpCKPFgwfD3LnhfNGikDSLSHoZJchm1hG4DTiaxuuNtcO7iIhIHmSy1XSdQYOSE+Rddmn82nXr4JVXYPhwGDCg9XGKlKpMSyzOB6qAEwkJ8hnAfwPPAe8Ch+UiOBEREWko0xILyHwli1tvhX79YL/9YOBAeOih1sUoUsoyTZC/CfwKuDc6f9Hdb3f3/YHZwKRcBCciIiINZTpJD5JXsvjgg9TXzJsHp54aRpDrHHEEzJ/f4hBFSlqmCfJ2wOvuXgtsAjrH7rsNOCaTJzGz28xsmZnNibVdaGaLzGxWdDs0dt85ZjbPzN42s4Nj7ePM7LXovuvMNCdXREQqR3NGkHfYIXH83nuprxk3DmprG7ZPmgSbNzc/PpFSl2mCvBzoEh1/COweu6830CnD57mD1KPN17j7mOj2GICZjQKOBXaJHnODmdXVOd8ITCZMGBzRyHOKiIiUpeaMIA8bljh+992G98+cCWvWpH7sO+/ABRc0Pz6RUpdpgjwdGBsd3w9cHI3u/hS4klCL3KRopYsVGb7m4cC97r7B3ecD84A9zWwA0NXdX3B3B+4CjsjwOUVEREpec0aQ4wny3LlhLeS4xx9PPp83D4YMSZzfdFNyQi5SCTJNkH8DvBUdXwL8i1CT/BvgPeDUVsZxhpm9GpVg9IjaBhFGq+ssjNoGRcf120VERCpCc0aQt90WttkmHH/8cXJd8dKlcNVVifOzzw4J9TOxhVtXroTnn299zCKlJKNl3tx9JjAzOl4LfNPMOgAd3L2RD2YydiNwMWFb64uBq4CTSL2cnKdpT8nMJhPKMejXrx/V1dWtDLd5ampq8v6ahaK+lif1tfxUSj+hfPu6YcP+1P06nD79adq29bR93XnnXZkxoxcAt9zyFpMmLQXgyit3ZPXqsJ5b9+4bqaqaQXV1KDo+7LCRPPLIQABuuOFDwqJVxaFcv6+pVFJfi4q75/UGDAHmNHUfcA5wTuy+J4G9gAHAW7H2bwM3Z/La48aN83ybNm1a3l+zUNTX8qS+lp9K6ad7efZ10yb3UCjhbua+ZUtoT9fXiy9OPOass0LbvHnubdsm2p98Mvkxf/5z4r6OHd2XLs1Nf1qiHL+vjamkvhYCMNNT5IwZlViY2TVmdncj991tZr9taYIe1RTX+QZQt8LFw8CxZtbBzIYSJuPNcPclwFozmxCtXnECoNUaRQrIPdQ2vvoqPPBAWEpqzZrkJaNEJDvqbxKSyTpOu8em1s+eHb5ecUVi5YoDDoCvfCX5MUcemahFXr8e7ruvxSGLlJxMa5C/DvyjkfueJMNJcmZ2D/ACsKOZLTSzk4EroiXbXgUmAmcDuPvrwBTgDeAJ4HQPy8xBqHn+A2Hi3rtAvSkGIpIPzz8Pxx0HW28NI0eGX8Lf/CZsvz106wZduoRf3j/+Mbz9dqGjFSkPmW4zHbfbbonjl14Kf8T+6U+JtvPPb/iYjh3hpz9NnF95JSxZ0rxYRUpVpgly/QlzcRlPknP3b7v7AHffyt0Hu/sf3f14d9/V3Xdz969HI8R111/q7sPcfUd3fzzWPtPdR0f3nRENkYtIHnz6afhlagZf+hL8+c9hdCmdq6+GUaPgJz8Jo1effpqfWEXKUXO2ma6z3XaJLaY//TT8EVv3/3C33WD//VM/7tvfhp49w/H774dd9hpbEk6knGSaIK8Ehjdy33BgbXbCEZFitWEDVFVB585wySXNf/yWLWG2/JgxYevb//f/sh2hSGVozhJvdczgtNNS33faaY2XafToAdddlzifNy98OpRqPWWRcpJpgjwV+KWZ9Ys3RufnAv/MdmAiUjy2bAlJ8dNPp77/iivg73+HQw8No1QzZ8Lq1cnrr8atWgVHHx1+Ke+1F0yfnrPQRcpOc5Z4izv++IYJ9YQJcPLJ6R/33e/CxRcntw0f3viufCLlINME+XzCTnpzzewvZnaFmf0ZeCdqPy9XAYpIYS1d2pHBg1OPGg8aBHfeGeoUDzsMHn0U5swJ29Z27QovvBCS6vXr4XvfS/3806eHJNkMfvtblV9IZaqtTb2JRyotGUGGsBbyEUckzgcNCvMI2mWw4Ou550KvXsltw4aFmEXKUUYJsrsvAPYA/kaYSPfD6OuDwB4edroTkTKzcSNccsnODSbmvPlm+EW+cCGccELjj+/TJ9QsdugAt90Wtq2dOrXxj3N/+tNQwrFkSWaJgkg5cA+rSIwcCaef3vT1LR1BBvj1r+GLXwyf9syYkdkKGABt2qROhkeOhD32CCvYiJSTTEeQcfcF7n5CNMmuvbsPdPf/cvf3cxmgiBTGBx+ExPb117sltU+dCjvt1LLnHDECDjwQHn44fNw7eHDq6wYODCPQZ54Jy5a17LVESsVbbyV2rrvxxsTSa41p6QgywNCh4VObRx8N/8+ao0eP8AlRfTNnhhVs/vd/9YetlI+ME2QRqRzLl8O++ya3HXJIGDE+8MDWP/9hh8Fdd8GHH4bkoE+fhtfU1MD110O/fqHeUXXKUq7eeiv5fMGC9Ne3ZgS5tXbZJZRMjRjR8L5f/Qp+97v8xiOSK41WHpnZbcDF7j4/Ok7H3b2JMn8RKQVr1sAxx4QR5Donnwy33pr5x7HNseOOYZT4L38Jk4FSeffdUKd83nlhBOwLXwjLxuU7ORDJhddfTz5/443GJ7hC60aQs6FDh1Au9e67cPbZYYJunR/8IJRVjRmT/7hEsildaf5E4Nro+AAg3Qcn+lBFpEycfTY89VTi/KtfXcxNNw3MSXIc953vhDVX//IXePBBuP/+htekmih40EFhdPvww6F//1DDLFJKPqy3y0BTI8gt2SgkF4YNgylTwv/dBx9MtI8dG1aq6dat0YeKFL1GSyzcfai7z46Oh0Tnjd12yF/IIpIrCxeGyXR1LrgAfvKTdzKa5Z4NZmEU+b774LXXQolFU6ZODTv1DR8edu477LCwxNyWLbmPVyQb1tbbSWDVqvTXF3oEOa5jx7C9/KmnJrefc05h4hHJloxqkM1sPzPr0sh9nc1sv+yGJSL5tmoV7LNP4nz4cLjoooKFw+jRYUZ/bW0orUhVp5zKo49C9+4hcRg1Knzke955IYmeOTOMdE2dqslEUjyamyB/9lnieOutsx5Oi/z+92E1izo33giLFxcuHpHWynRcaBqwFzAjxX07Rfe3zVZQIpJf7mHk9v3YmjTnFcnq5m3ahE0KLr44TNw7++xQfjF6dJhR37lzGPmub/PmsBzdm28m2q6+Ov1rjR+/G9ttF1YUGDUKtt02JNjxX/wi2VY/QV65Mv318bXCiyVBNgsTabt1C/9PIZQ9vfRSYeMSaalMV7FIV33YAWhiURoRKWb33AOPPZY4v+ACOPHEwsXTmC5dwmTBFStCErtiRajfXLIETjqp9c8/c2ZPHngAPvkkPP+f/wx77hl++XfrFj421sizZFtzR5DjCXKnTlkPp8XatIHLL0+cz5wJ//lP4eIRaY1GE2QzG2JmB5jZAVHT+Lrz2O2rwI+BDxp7HhEpbmvWwE9+kjgfNgx+8YvCxdMS/fvDH/8YkteXX4ZLLw3rKGfTmjXhl//JJytJluwqhxKLOqeemhzTQQcVLhaR1kg3gnwiMBX4J2GVit9F51Nj7X8Hvgb8JrdhikiuXHwxn++U17cvvPJKcY1KNdcXvhC2xV26NNRAuofbypVh0t8TT4S1Wi+4oGUrANx+exgpu/LK7Mculak1JRbF9n+1TZvk1WZWrAh/tIqUmnQ1yHcA1YTyin8BpwNv1LtmA/COu6/IRXAikjvucPTRYcWIOldfnf2R10Lp1Ck5eejePbGN78EHh68XXRRKNLbeGrbaCm6//T9MnDiWUaOgXbvwh8PRR8NzzzV8/p/9LIwqn3++1mOW1lmzJvm8OSUWxTaCDHDWWfCjHyXOTzsNXnyxcPGItES6Zd7ed/engX8DRwLPufvT9W7TlRyLlKabb05OjvfdN6xnWmm23RZ69Qp/GOy++2p2243Pl7UbMACefTZMOvrZzxo+9pJL4Ic/zGu4UmZqa5MTXmh6BLmYSywA2rYN8xrqvPRS00m/SLFpcpKeu28C/gr0zn04IpIPK1bAT3+a3Pb73+dmp7xy0Lkz/OY3Yf3ZX/0q+b4bbwyT+URaom7Fh7g1a9Kv413MJRZ1jj0WRo4Mx+7w5JOFjUekuTJdxeI9oG8uAxGR/Ln55uRfzG+/DbvuWrh4SkX79qGkov4GJscdB/PnFyYmKW31648hJJSrVzf+mGIfQa5z9NGJ45NOSt1XkWKVaYJ8BfBLM8twqX4RKVZTpoRJbHVuvDEx0iOZOf30UHoRd9xxhYlFSltjSWO6koRSGEEGOOOMRHyffhrWRdYKMFIqMk2QDwB6AvPNbKqZ3W1md8Vud+YwRhHJkqVLw4YgdUaPLs71jkvBPvskrx39/PNh7WSR5mhtglzMI8j9+sGZZybOp00LO12KlIJME+R9gE3Ax8Cw6HzfercmmdltZrbMzObE2nqa2T/NbG70tUfsvnPMbJ6ZvW1mB8fax5nZa9F915mpclIkE7feGnaYq3PjjcU9AlXsDjkkeeT49NOT/31FmtJYgpxuol6plFhAWEZyzz0T54Xcvl6kOTJKkN19aBO3HTJ8vTuASfXafgE85e4jgKeic8xsFHAssEv0mBvMrG476xuBycCI6Fb/OUWknieeCGv/1vnRj8IoqLTOr38dJvFB2Pq6fumFSDotGUGOJ89dumQ1nKxr3x7+/vewjCKE3fUefriwMYlkItMR5Kxw92eA+svCHQ7UlWjcCRwRa7/X3Te4+3xgHrCnmQ0Aurr7C+7uwF2xx4hICo8+Cocdljjv3z8kdtJ6gwcnjyIfcEDj14rU19wRZHd4//3E+XbbZT+mbOvbFw49NHF++OGwbFnh4hHJRLqNQpKYWWfgZGA/oBcw2d3nmtmxwCx3f6uFMfRz9yUA7r7EzOpWyxgETI9dtzBq2xQd129vLO7JhNFm+vXrR3V1dQvDbJmampq8v2ahqK/FaenSjpx44h7U1oYPYLp23cQvfjGH559PM00+ppT62lot7WuvXn2BUZ+f33zzTHbcMcX6XUVC39Pi8fLLA4GGs2T/8595VFcvbNC+cuVWfPbZlwDo3Hkzs2Y99/nyjMXc12OP7chDD034/HyffdZy880vt3hpyWLua7ZVUl+LSUYJspltS9hVbzDwFjAa2Ca6eyJwEPDfWY4t1X8bT9OekrvfAtwCMH78eK+qqspKcJmqrq4m369ZKOpr8fnoI5g8GTZuDOddu8KLL27FTjuNzfg5SqWv2dDSvu65J1x2WeJ88+bxFPM/mb6nxaOxHeb69h1OVdXwBu0zZiSOhw1rx8SJVZ+fF3tfZ8yAa64Jx3PnbsPmzVV8+cste65i72s2VVJfi0mmJRZXEbaVHgGMIzlJfZowqtxSH0VlE0Rf6z54WQhsG7tuMLA4ah+col1EYtzhv/8b5s4N523bwiOPwE47FTaucrT11vDb3ybOp09v/FqRuHiJRbyeuP7203UWLEgcDxmSi4hy5/LLk7eyj/+fESk2mSbIXwb+190/oOFo7SLSlDhk4GGgbqGpE4GHYu3HmlkHMxtKSM5nROUYa81sQrR6xQmxx4hI5Fe/CglxnWuuCdtJS27EJzzedx+8807hYpHSEU+QBw5MHJdjgty+Pbz6auL8H/9I3u5epJhkmiC3BxrbA6cboS64SWZ2D/ACsKOZLTSzk4HLgS+b2VxCIn45gLu/DkwB3gCeAE5399roqU4F/kCYuPcu8HiG/RCpCE8/DRdemDg/6KCwaL/kzp57wpgx4Xj9+uTNWEQaE0+EBw1K3R4XT5CHDs1JSDm1/fbwxS8mzk84IXnSoUixyHSS3qvANwmJan2HAC9n8iTu/u1G7jqwkesvBS5N0T6TUActIvU89xxJdX3jxsEDD9DiyTCSGTO49lrYf/9w/uijYTvvYl+GSworPoIcT5AbW92ilEeQ61xxReL/yWefwcEHw5tv6j1KikumI8hXAieb2a0k6o1HmdlFhJUtrsxFcCLSPOvXhzKKTdFnOlttFUorttkm/eMkO/bbL+xOCOF7oV3DpCmNJciNjSC//XbiuBRHkCH8P7nqqsT522/DbbcVLh6RVDLdKOQB4DTgW8DUqPku4IfAGe6eamRZRPJoxYrkCTDt2sFTT6nuON++9a3EseorpSnNSZDXrYP33gvHbduW9oTbI49MPr/sMqitTX2tSCE0miBH20J/vjqFu99EmIx3MHAcobRicLSMmogU0OLF0KtXYuQY4PjjlRwXwlFHJY4ffTQkNSKNaU6C/MYbieORI6FDh9zFlWtDhsAzzyTO33tPq1pIcUk3gnwMMM3M5pvZRWY2zN3XuftUd/+Luz/p7o1N3BORPHnrreRfrBC2Pr744sLEU+lGjQo3CPWV06YVNh4pbs1JkN+Kbcc1alTD+0vNvvsmTya+5BLtsCfFI12C3I+w+ccC4DzgHTN7zsz+x8y65SM4EWmce/hYcuedk9tHjYLVqxsmzZI/X/1q4vhxrbEjaaSbpOf1FlWdNy9xPGJEbuPKl1/+MpHs19SE5SlFikGjCbK717j77e4+ERgCnA/0BG4GlpjZPWZ2iJllOtFPRLLk3XfDShW//GVy+8iRMHNmqE+Uwpk0KXH8+OMNEx2ROvEEuXfvMLEWQrnUhg3J19Zt+gPlkyC3axc2EKnz+99rcqsUh0wn6X3o7pe5+yhgAnAbYWm2R4BFZqbKIZE8WbUKhg8PE/DqtG0LZ58dPoLt1KlgoUlkn30Sy7vNn5+8NJdInQ0bEvMG2rULNcXxibb1yyzKMUEGOOwwkrZmP+yw8Ie+SCE1e/TX3We4+xmECXvXAH2Bs7MdmIg09Je/QI8eyW2nnBJqXa++WuuIFov27ZM3Q/jPfwoXixSvVasSx926hf+/jSXI7skJ8vDhOQ8vb8zgjjuSP/mKfwojUgjNTpDNbLiZ/Qp4m5AYryWMKItIDmzYADffDN27w3e/m3zf3nvDjTcmPpaV4lG3qx7Aiy8WLAwpYitXJo7r/vBtLEH+5JMwtwDCpxP9++c+vnzafnv42tcS58uXJ6+VLJJvGSXIZtbTzE41sxcIifG5wDuE5d76u/v/5DBGkYr1/PPQsWMYJa775VjntNPgb38rSFiSgfgI8h13wMaNBQtFilQ8Qe7ZM3yNJ8jx+uT4ChbDh5fnp0UPPBBKTer87//CokWFi0cqW7p1kLcys2+Y2YPAYuD3wDbAOcB27j7J3e9x9/V5ilWkYnz6aUiKv/Sl1Pe/9FKYzNKnT37jksx9/evQt284XrYseQ1bEQib+9SpG0GO73oZH0H+178Sx/FPJ8qJGbzwQuJ83Tr4xjf0x2Uq7vDqq/DTn4ZylB/8IPwxcdJJidH4KVMaTvSUzKUbQV4K3AfsA9wK7Onuo939CndfnJfoRCrQww9Dv36hrKK+UaPCqPL48fmPS5qnQweYMCFxrgRZ6muqxCL+qVF1deL44INzGlZBjR8PU6cmzl96CW66qXDxFIs334QLLgh/RJiFkfbddw+bqzz5JFx3HQweDLffDh98AI88AsccEz6BPOaY5E1ZJDPpEuRngaOAge5+prtrTqlIDrz1Fvz972GLYjM4/PCwHmh9L70Er78Oe+2V/xilZXbZJXH86quFi0OKU6oEOT4Jt24SnzvMnp1oL/f3gAMPhEsvTZz/+MdhInKlqa2FBx+Es84aw6hRyZs/bdmS+fNMmQL77x9+v4wcCT/7GVx7bfIkUWmoXWN3uPsReYyjLD3yCMye3Yt33gk/zJ06wUEHhcXgN20KI0oDB4aPydetg1deCaNOw4aFN8mNG6FNmzAjvjnWrw9/Nbon1l+dPz+MRvTqFXZZcw9LT/XvH9rMwmM++wy23jrr/xSSwhNPhO2gP/mk8Wsuuwy++U0YMCD5o1cpDV/4QuL4r3+FX/+6PGtHpWVS1SDHE+S6EozFixPXdu0K222Xn/gK6fvfDyOmtbWweTP85jfJu+6Vu+efj5fYdc/a886dC1deGY5/+MPw9bzzwrr6w4aFnGPAgNa/zocfhlxkyZKQ2zzzTBjhfuedkIMsWwZHHRWue/bZUFd/yCHh05FiKR1sNEGW1lm4sG5G7q5Zeb6BA0M9Wv/+oT61Tx/YdVc49NAwMnXttdn7C3vQoBD7qlVw7LFhRFOyY8OGsKLBgQeGN/10nn++/EeKyt1XvxqW71q9OvxBumABDB1a6KikWMRH8Lp3D1/rEmVIJMjxTx923bUy/sjq1Sv8HqqbiHzRRWGAaZ99ChpWzm3eDHvsAbNmpb5/551h7Fg466yQ0C5fHhLa+fND/fZ++4VrrrkGXnst/IFx993pX/OSS8Ktvt12CwntTjvBjjuGwbquXWHIkPBe9sEH8PTT4Y+6999vfl+fey5xPGNGWMa0Tx9YujQk6oVmXkFbPI0fP95n5mn18RNPhLvuqoB3MREREZEs+Pa3nb/8Jb+vaWYvu3uDmT1FkKOXp6OOKnQEIiIiIqXjsMMKHUGCEuQc+drXYNMm53e/e5kbbnAWLHBefNEZNswZONC54QZn4ULnkUecr3zFGTfO+cEPnMcec6ZNc1ascB580AGnZ0/nyiud3/0unNfdDjgg+XzQIGeffcJxmzaJ9r33dmbNcpYudb73PeeMM5yf/9w57bQQ0/TpzqJFzk9/Gq7v2tU5/XRnhx2Sn7+p2267rWTKFKe21nEv79u0adOavGb27PT/Xvvs42zYUPi+ZKOv5XLLRV8vuSTxPT/jjML3Ud/T4rkdemjiZ+Phh0PbM88k2vbaK7Tttlui7ZlnSrOvrbnddVfye+ceezhTp1YXPK5s3D74wGnXLvXviBNOcLZsyf339bPPnPPOS7zuuHHOUUeF3GPiRGfyZOfIIxP377uvc8UVzq23Ohde6Jx9tnPIIc63vuX87W/Oz37mXH99eM5DD3WmTg35zpYt4ef8mWecN990Vq92Nm5MjuU738l6OtZiqkHOoXbtYPToNZ/vMb/99jBvXvI1gwaFOsVUjjgiMcmuzte/Dh99FGqU3MN+9TU1YR/7eF2ae+o6tdvS7Hl4xRWJSURt2oSJhU8+GWqc2rSB++8PC7k35tVXu3P00eH46KPhd79LrANbabZsCd+/+vr1g0cfDXVcI0bkPSwpgHHjEsdPPlm4OKT4xFer6dIlfI3XIC9fHiZ0v/lmom3X7ExrKSnHHx8mkdf9fnnpJXj00QEceGBh42qtV15Jfn+os+OOYam7wYPzE0fHjmGFjPgqGal8/HGolW9q59Z085biuyUWO40gl5jttgvJMYREdo89YOLEhslwSydxtG2bKI5v0ybMKj3uOPjOd0KCvH59+I/7s5+lf54pU8LEwjPPbHoyWjn517/CHxkTJoRJE3HXXRdm9I4bp+S4klRVJVaGmTtXO4NJQnynvLpVavr1S7R99FGYaLVpUzjfdtvEZL5K861vwTnnJM7/+MehfPRR4eJprRkzUifHt9wSlvPMV3LcHH36NJ0cl5OiSZDNbIGZvWZms8xsZtTW08z+aWZzo689YtefY2bzzOxtMyvjZdOLS4cOYQWG3/wmjJJ+/HH4j57qL8baWrj++vAfKt3Idbl44IHwb3PuuWGEo86oUeHf6swzK2P2uSTr2DF5ubfGZqdL5UmVIPfqlUhCVq9O3iRj773zF1sxOv/8sIICwJo1W9G/PyWXJNfWwk9+krwVfZ1nnoH/+Z8wUCWFVzQJcmSiu4/xxGzCXwBPufsI4KnoHDMbBRwL7AJMAm4wM/1I5ZkZ9O4dRrH/9jeYNq2aG29Mfe3JJ8M99+Q1vLxZsSKMbnzzm8ntbdqEZfJeekmJcaXbfffEcXzDB6lsqRJks+R1aG+9NXF80kn5iatYdeoUSgHjmvo0s5gsXBiWYbvqquT2ww4LZZH77luYuCS1YkuQ6zscuDM6vhM4ItZ+r7tvcPf5wDxgz/yHJ/Wdckr4j/7eew3v+853wjqW8e1TS93ateFjsvvuS24/8UR4++3wR4E2XpExYxLHGkGWOqkSZAjlafV16hR2Q6t0X/968vndd5dGbf/06aFE5vnnk9vPPz/spCrFp2jWQTaz+cBKwjTJm939FjNb5e7dY9esdPceZnY9MN3d/xS1/xF43N3vS/G8k4HJAP369Rt377335qE3CTU1NXSpm31R5ur3dcaMnlxwwS5s2JA8uL/rrqu45prZtG1bHD97LVFTU8Pq1b057rgJSe2DB3/KlVfOpn//DQWKLPsq+Wc4W95+extOOSUUHA4e/Cl33z0j66/RHPqeFl5tLRx0UNXn5089Vf35/I8LLtiFZ59N3k5s/PgVXHll+v3Ki7Wv2bZoUccG770nnriA//qvBYUJKA13uP764TzwQHJR8bhxKzjrrHlst92nTT5HpXxfC2XixIkp10Eu+BIndTdgYPS1LzAb2A9YVe+aldHX3wPHxdr/CHyzqdcYN26c59u0adPy/pqF0lhfTz/dPbHxdbi1bes+Z05+48umn/zkrQZ9GjvWfdmyQkeWffoZbr1PPw0/8+Bu5r52bU5eJmP6nhbe6tWJ947OnZPvO//8hu+Zv/51089ZrH3Nhbvvnu5duiT/G/3734WOKll1dcPvI7gPHuy+cWPmz1NJ39dCAGZ6ipyxaEos3H1x9HUZ8CChZOIjMxsAEH1dFl2+ENg29vDBwOL8RSvNcf318NRTyW21tTB6dNNbYBabf/wj1Aj+9rc7JrVvvXXYQrpY9pCX4tKpU1i6CcKvyNdeK2w8UniNlVcAjG84lsXBmoqeZPDgz3jhheS2L30psTV1oV10EZ8v8Ro3aVIoQayk1SBKVVEkyGbW2cy2qTsGvgLMAR4GTowuOxF4KDp+GDjWzDqY2VBgBFDYzywlrQMOgA8/bPiL4IQTwrJotbWFias5Pv644UQ8CH1aulRveJKe6pAlLl2CvPfeYfWTOqNHJ//8SDB6dPIkRoBvfCMst1kINTXwy1+GgZILL2x4/09+EhJ4/a4oDUWRIAP9gOfMbDYh0X3U3Z8ALge+bGZzgS9H57j768AU4A3gCeB0dy+BFKuyDR4M//lPwyTz3HPhv/6r4aYoxeStt8KmJ/GF/SGsDb16dcNfcCL1KUGWuHQJcu/e8Ic/QLduYc30u+7SSjiNOfnksO583IEHNvzUMtduvTV8Hy+7DD75JPm+738/rIt/5ZVhqVQpDUWRILv7e+6+e3Tbxd0vjdqXu/uB7j4i+roi9phL3X2Yu+/o7o8XLnppjmHDwiYiw4Ylt//pT2FZtFtuKb5Euboadt45uW3s2JW89hoceaR+cUlmlCBLXLoEGeC73w2fWr3zDowdm7+4So0ZPP10w7WDDzoojNguWZK713YPm3rssQdMntwwrqqqMLhy002J9ZuldBRFgiyVpU2bsKPYj37U8L7vfz98RFYMu425h2XaJk5Mbh85Ei69dA6jRxcmLilN8bWQX3utNMqKJHfin0Y19gmUPorPTPv24ZO8+nXaV10Vlsx75pnsvt5TT4VBnjZtQpnHzJnJ9w8aFH6HTZuWmHsgpUcJshSEWXjzWrUKzjor+b6HHgrlGD/8IWzcmP/Y3MNmDocdFtZujps4MYwIdOqk7Eaap2/fxAYQn30W/kiUyhUfQdYKXq3XuTM88QRcd13D+/bfP/wb//OfsG5d857XHZYtC7+rHnoIvvzlMDqdaq1/gCOOgDlzkjd7kdLUrtABSGXr1g2uvTZsVX3ggcn3XXttuB13XHjT69Ej9XNkQ21tWIWia1c45xx45JGG14wZE94gVVIhLTVmTOIj31mzYKedChmNFFJTJRbSMmeeCbvuGn53xFe0WLcOvvKVcLzPPmHUuXPnUO89YkT4ZHDhQhg+HBYsCEnxe++F8r+m9OoVRq9POy2spCHlQQmyFIUDDggfSY0bF1aEiPvTn8Lt6KPDRIiuXTN7zo8/DtfWnxRRWwubNsGzz8K774aPyx57DD5Ns177V74SJuRppEdaY8wYeDyaMfHqq2ErcqlMSpBzp6oqjBoffHAYNa7vueda9/xt2oTv2erVYYL5TTdp8l05UoIsRWPgQHjlFbj99rB1aP26sSlTwm3nncNf6XvvHSY+zJ0bapd33jncv3JleHOsm+zXtWtYe3L//cPalMuWNXjplKqq4Hvfg29/W7WAkh3xEWOVWFQ2Jci5ZRbWrb/6avjxj7P3vAcfHJ5z1KjsPacUJyXIUlQGDAjLvp17bhhR3nffsDxO3Jtvhtsf/tCwfdddGz7nmjWJ5DpTY8fC1KkNZ0aLtMaIEYljJciVLf5JmTYYyp0f/SjcPv00rDRRN6Fu4cLM65HPPx922w169gyfdkplUIIsRWvQoLBu8sknh/KGXLvuOli+HPr1C0ssKTmWbKufIG/ZEj6ulcqzOLb366BBhYujUmy9deP1xA89FD61PPLIMAEPwu8es7D6jOadVCYlyFLUunWD++4LdcMzZ4aJeo8/HkaY09UMt2kTPs5+//3EKEHd0jsQakGvuip8NQvP37t3rnsjla5Xr/AH2EcfhZ/f115LXv5NKkd8KcuBAwsXh4RJ4ocfntymtadFCbKUhLZt4YtfDMcjR8Ipp4Ql4Lp0CRPuVq4Mt222gf79E6O/n34aJlLULbmzYgWsX69fSFIYZqEWvq7c55lnlCBXKo0gixQ3JchSkjp0SMwabt8+jMr169fwuq23Drc6PXvmJz6RxkyYkEiQX3+9sLFIYWzenJgsbJb6vUtECkvVbyIieTRyZOJYE/Uq0/LlieNevaCdhqpEio4SZBGRPBo+PHE8b17h4pDC+eSTxLHmPogUJyXIIiJ5NHRoYl3tDz5QklyJ4gmylngTKU5KkEVE8qh9+7DZQJ2//rVwsUhhfPxx4lgjyCLFSQmyiEieHX104vippwoXhxTGnDmJYyXIIsVJCbKISJ7Fd+N6/vmwqoFUjhtuSBwrQRYpTkqQRUTybNCgxFrcGzbAu+8WNh7Jn88+Sy6xOOSQwsUiIo1TgiwiUgCjRyeOtR5y5YhP0OvcGfbdt3CxiEjjlCCLiBTArrsmjjVRr3LEE+QddihcHCKSnhJkEZECOOaYxPGUKfDkk4WLRfInXl6hJd5EildJ799jZpOAa4G2wB/c/fIChyQikpE99oDDD4eHHgrn998fln/77DN45RXYuBFWroTddw+7rS1dCt27Q//+sHYtzJoVRiNHjIBddoEXXoD33w/P268f/Oc/0K1bWFZu0aJQ0tGmTVh3ee3admzYELZr37AB/vIXGDYM1q0Lr/XhhyGO7baDMWPg2WfDDoBz50LbtuF46NDk/mzZAi+/HJ6zW7cQw7x5Id7eveGjj2D16hDv3Llh2/fVq8OKDl27hv7tthvU1ob2wYND7H/7W3j+I48Mtdrjx4fz5cvBPbS1bRv61r9/2Fq+rq577Vr497/707dvuGbx4tC/Hj3g009DLG3awPz5sH596NecOeG1//1vGDs2bAn95puhFKKmJlw3dmx43Jo18N574d9/06awrvXw4aEvvXuH78+aNTBqVOLf6f33E8dKkEWKV8kmyGbWFvg98GVgIfCSmT3s7m8UNjIRkcz86EeJBPnWW8MtP/bJ1wtlzRlntPSRO3HlldmMpGXGjQvJcnx7ca1gIVK8SjZBBvYE5rn7ewBmdi9wOKAEWURKwoQJYSR1xYpCRyK59vLLDdt23z3/cYhIZszdCx1Di5jZUcAkd//v6Px44Ivufka96yYDkwH69es37t57781rnDU1NXTp0iWvr1ko6mt5Ul9z6/77B3H99SPy8lpmjrvl5bUkvbFjV3L55a/Rvv2WrD2n/q+Wp0rqayFMnDjxZXcfX7+9lEeQU73LN8j23f0W4BaA8ePHe1VVVY7DSlZdXU2+X7NQ1NfypL7mVlUV7L03zJgBb70VPobfsiWsctG/f7h/6FD43e9CbWxVVaiZ3X77MLFv0aJQP3zooeHr3XeH2tvjjoPZs2GbbcJtu+0AjM6d4a9/fYH33tuLmhqi81D/+8Mfhhre2bNDjfMxx4RYOnQI53vtFTY1uf12+PKXQ43ws8+G0pAFC+A73wmvs/XWoZ7aDCZPDo89/vjQ365d4dxzoVMnePrpcD5vXhhN79QJrr0WJk4M9dcA3/1uiG/jxlCz/bOfQceO8LWvhTrhrbcOz73ttqFW+LnnQtvSpXDLLTB+/AcMH74d77wDV1wRnvfqq2HJklCvvc02oV67R4/wHMOGhRrw118P/96nnBLKI2bPhn/8A9q1C7XMffqEEokDDgjHGzaE558zB77+9fD869aFrx06hPifeQaOOCL8m4wa1QPYL6s/S/q/Wp4qqa/FpJRHkPcCLnT3g6PzcwDc/deNPWb8+PE+c+bMPEUYVNIPtvpantTX8lMp/QT1tVypr5ItZpZyBLmUl3l7CRhhZkPNrD1wLPBwgWMSERERkRJXsiUW7r7ZzM4AniQs83abu2s/KhERERFplZJNkAHc/THgsULHISIiIiLlo5RLLEREREREsk4JsoiIiIhIjBJkEREREZEYJcgiIiIiIjFKkEVEREREYkp2o5CWMLOPgffz/LK9gU/y/JqFor6WJ/W1/FRKP0F9LVfqq2TL9u7ep35jRSXIhWBmM1Pt0FKO1NfypL6Wn0rpJ6iv5Up9lVxTiYWIiIiISIwSZBERERGRGCXIuXdLoQPII/W1PKmv5adS+gnqa7lSXyWnVIMsIiIiIhKjEWQRERERkRglyCIiIiIiMUqQs8TMJpnZ22Y2z8x+keJ+M7ProvtfNbMvFCLObMigr9+N+viqmT1vZrsXIs7Waqqfsev2MLNaMzsqn/FlUyZ9NbMqM5tlZq+b2dP5jjFbMvj57WZmfzez2VFfv1eIOLPBzG4zs2VmNqeR+8vifSmDfpbFexI03dfYdeXwvtRkX8vofampn+GyeV8qGe6uWytvQFvgXWAHoD0wGxhV75pDgccBAyYALxY67hz2dW+gR3R8SCn2NZN+xq77F/AYcFSh487h97Q78AawXXTet9Bx57Cv5wK/iY77ACuA9oWOvYX93Q/4AjCnkfvL5X2pqX6W/HtSpn2Nrin596UMv69l8b6UYV/L5n2pVG4aQc6OPYF57v6eu28E7gUOr3fN4cBdHkwHupvZgHwHmgVN9tXdn3f3ldHpdGBwnmPMhky+pwBnAvcDy/IZXJZl0tfvAA+4+wcA7l6q/c2krw5sY2YGdCH8Itqc3zCzw92fIcTfmLJ4X2qqn2XyngRk9D2F8nhfyqSv5fK+lElfy+Z9qVQoQc6OQcCHsfOFUVtzrykFze3HyYQRqlLTZD/NbBDwDeCmPMaVC5l8T0cCPcys2sxeNrMT8hZddmXS1+uBnYHFwGvAD9x9S37Cy7tyeV9qjlJ9T8pIGb0vZaJc3pcyUUnvS0WhXaEDKBOWoq3++nmZXFMKMu6HmU0k/DLaJ6cR5UYm/fw/4OfuXhv+qC9ZmfS1HTAOOBDoBLxgZtPd/Z1cB5dlmfT1YGAWcAAwDPinmT3r7mtyHFshlMv7UkZK/D0pU/9HebwvZaJc3pcyUUnvS0VBCXJ2LAS2jZ0PJvyV19xrSkFG/TCz3YA/AIe4+/I8xZZNmfRzPHBv9EuoN3ComW1297/lJcLsyfTn9xN3XwesM7NngN2BUvtFlElfvwdc7u4OzDOz+cBOwIz8hJhX5fK+1KQyeE/KVLm8L2WiXN6XMlFJ70tFQSUW2fESMMLMhppZe+BY4OF61zwMnBDNGp8ArHb3JfkONAua7KuZbQc8ABxfwn/JN9lPdx/q7kPcfQhwH3Baif4SyuTn9yFgXzNrZ2ZbA18E3sxznNmQSV8/IIxIYWb9gB2B9/IaZf6Uy/tSWmXynpSRMnpfykS5vC9lopLel4qCRpCzwN03m9kZwJOE2cO3ufvrZnZKdP9NhNnEhwLzgE8Jfw2WnAz7egHQC7ghGsXY7O7jCxVzS2TYz7KQSV/d/U0zewJ4FdgC/MHd0y4zVYwy/L5eDNxhZq8RShB+7u6fFCzoVjCze4AqoLeZLQT+F9gKyut9KYN+lvx7Up0M+lo2muprubwvQUbf17J5XyoV2mpaRERERCRGJRYiIiIiIjFKkEVEREREYpQgi4iIiIjEKEEWEREREYlRgiwiIiIiEqMEWUQkz8zMM7gtMLMh0fF/FUHMQ+rFV9WMx54Xe9zC3EUpIpIdWgdZRCT/9qp3/iAwG7gw1rYBWBJd+25+wsrIJcCjwBvNeMztwFTgfMJOZyIiRU0JsohInrn79Pi5mW0gbJk7PcXlqdoK6d1G4myUuy8CFpnZxzmKSUQkq1RiISJSpFKVWJjZHWa20MzGm9nzZvaZmb1tZl+N7v9RVJ6xxsweMrM+9Z6znZmdY2ZvmdkGM1tsZleZWcdWxHlwFMtqM6uJ4rmgxR0XESkwjSCLiJSersBdwG+BxcAvgfvN7PfASOB0oB/wf8DvgaNjj/0T8DXgN8DzwM6EbWyHAN9sbiBmtgPwMHAf8CtgIzAC2KHZvRIRKRJKkEVESs82wCnu/gyAmS0m1DAfBoxy99qofTRwppm1dfdaM9sXOAY40d3vip5rqpmtAP5kZmPcfVYzY/kC0B441d3XRG3/ak3nREQKTSUWIiKlZ11dchx5K/o6tS45jrW3AwZE55MII7z3R6UW7cysHfCP6P79WhDLLGATcK+ZHWVmfVvwHCIiRUUJsohI6VkVP3H3jdHhynrX1bXX1Rf3JYz21hCS2rrbsuj+Xs0NxN3nAQcTfp/cDSw1sxfNbP/mPpeISLFQiYWISOVYDqwH9m3k/sUteVJ3nwZMM7MOwJcItciPmtkQd/+kRZGKiBSQEmQRkcrxBPBzoJu7P5XtJ3f3DcC/zKwL8BAwFFCCLCIlRwmyiEiFcPdqM7sHuM/MrgZmAFsIK1gcCvzc3d9pznOa2SmE2uXHgA+B3sA5hNHoOdmLXkQkf5Qgi4hUluOAM4GTCMvDbQAWAE8CH7Xg+WYDhwC/JtQ4rwCeA77r7p9lIV4Rkbwzdy90DCIiUuTMbAgwHziZsAZzrWf4C8TMDGgL/BE40N0H5ypOEZFs0CoWIiLSHH8krHzRnFUqfhk95oScRCQikmUaQRYRkSaZWXtgt1jT2+6+NsPHDgAGRacb3f3VbMcnIpJNSpBFRERERGJUYiEiIiIiEqMEWUREREQkRgmyiIiIiEiMEmQRERERkRglyCIiIiIiMf8fXq/B/JCC+yEAAAAASUVORK5CYII=\n", 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