{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"# Bisection Algorithm\n",
"\n",
"### Modules - Root Finding\n",
"\n",
"By Magnus A. Gjennestad, Vegard Hagen, Aksel Kvaal, Morten Vassvik, Trygve B. Wiig and Peter Berg\n",
"\n",
"Last edited: February 23th 2018 \n",
"___"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### The Task"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"How can we find the root $x_0$ of a function $f(x)$, i.e.\n",
"$$f(x_0) = 0,$$\n",
"if we cannot determine it with pencil and paper.\n",
"\n",
"Note that there could be more than one root and we would like to have some control over which root is computed."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Example"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"We would like to know where the function $g(x) = x$ intersects the function $h(x) = e^{-x}$. In other words, we need to solve\n",
"$$x = e^{-x}.$$\n",
"This is equivalent to finding the root of $f(x) = e^{-x} − x = 0:$\n",
"$$f(x_0)=e^{−x_0} −x_0 =0.$$\n",
"Let us plot the two functions g and h to gauge whether there might be a root and, if so, where approximately."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"%matplotlib inline\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"# Set common figure parameters\n",
"newparams = {'figure.figsize': (16, 6), 'axes.grid': True,\n",
" 'lines.linewidth': 1.5, 'lines.markersize': 10,\n",
" 'font.size': 14}\n",
"plt.rcParams.update(newparams)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
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FGA88+3l44RaoOXLmZ0VEREREpNdYtGgRX//61yktLe3wM+Hh4TzyyCNdWNWHaaS5HRpp7iYtTbD6F7DifyBqICz+f3De1eqwLSIiIiLSqjePNHc1jTRL8AsNh3nfh6+sdLapeuFmeO5GOF7mdmUiIiIiIiIKzdJDJI+DW99yOmzveBMenQFbngXNhBARERERERcpNEvPERrmdNj+6mpnX+eXvgJ/vR6O7ne7MhERERER6aMUmqXnSRrr7Ou88EHYuxoenQnv/A78LW5XJiIiIiLiCvWiOnuB+mem0Cw9U0gozLoD7ngHRlwAy34Af7wUyt53uzIRERERkW4VGhpKU1OT22UEnbq6OsLDwzv9HoVm6dniU+HGv8Nn/gBVe+D3F0HOT6Cp3u3KRERERES6RVxcHGVlZfj9frdLCQrWWmprazlw4AAej6fT79OWU+3QllM9UM0ReP0/YOuzkDAGrvgVjJzjdlUiIiIiIl3K7/dTWlpKTU2N26UEjfDwcDweDwMGDGj3no5uOaXQ3A6F5h5sZw7861twdB+c/2W45D6Ijne7KhERERERCSLap1l6rzELnLXOs74Bm/8Cv54OW5/X9lQiIiIiIhJwCs0SnCJiYOEDcPtyiBsBL94GT30aKna6XZmIiIiIiPQiCs0S3IZMglvfhMsfhoP58NtZkPegGoWJiIiIiEhAKDRL8AsJhen/Bt9YD5lXwoqfwm9nw+7lblcmIiIiIiJBTqFZeo/YZLj2j3Dji2D9znTtv98Gx8vcrkxERERERIKUQrP0PmMWwB1rYd4PYPvL8Otp8M5voaXZ7cpERERERCTIBE1oNsbcbYxZb4w5ZowpN8a8YowZ34HnJhhjVhhj6owxB4wx9xpjTHfULC4Kj4as/4CvrYVh02HZD+GxeVCy1u3KREREREQkiARNaAbmA78BZgPZQDPwljFmUHsPGGMGAG8CZcB04C7ge8C3u7pY6SESx8CNf4cbnob6anhiEbz0VTjhc7syEREREREJAsYG6d62xpj+QDVwlbX2lXbu+RrwMyDZWlvXeu4e4GvAMPsJX37atGl2w4YNgS9c3NNYA6sehjW/ckais++BabdCaJjblYmIiIiISDczxmy01k47033BNNL8UbE49Vd9wj2zgFUfBOZWrwMpwMiuK016pIgYWHAv3PEODJsGr32/dcr2225XJiIiIiLSq1TXNrldQsAEc2j+JZAPfNIi1cE4U7NPVXbKNemLEsc4Hbav/wvUHYUnLoMXboHqUrcrExEREREJSi1+y6Z9VTz8RhGLf7mKGUveoraxdzTiDcp5qcaYnwNzgbnW2pYAvvd24HaAESNGBOq10hMZA+OuhDEXw5r/hTW/hKLXYO63YfY3nOnbIiIiIiLSrmP1TawsLie30MeKonKO1DQSGmKYOiKeb108lmZ/cC4F/qigW9NsjPkF8Fkgy1pbeIZ7nwISrLWXn3JuOrAOGG2t3dPes1rT3MdUlcCb/wnb/wFxI+DSByDzCidci4iIiIgI1lp2ldeQV+gjp7CMDXuraPZb4vqFM39sElleD/PGJhHXL8LtUjuko2uag2qk2RjzS+AGOhCYW60FfmaMibLW1reeuwQ4COztmiolKMWnwvVPwZ6V8NoP4PkvwqiLYNHPIHmc29WJiIiIiLiiobmFdXsqySnwkVfko+RILQDewbHcdtFoFng9TB4eR1hoMK/8/WRBM9JsjHkU+CJwFbD9lEsnrLUnWu95EJhhrV3Q+vtAoAhYDtwPjAWeBO6z1j78SZ+nkeY+rKUZNj4BufdDw3GYdgvMvxtiEtyuTERERESky/mO1ZNX5CO30MfqHRXUNLYQGRbC7LQEsr0esjOTGRoX/MsZOzrSHEyhub1C77PW/nfrPU8C8621I095bgLwKDADp9P274Aff9J2U6DQLEBtJeQ9ABuegIj+MO/7MON2CAuO6SYiIiIiIh3h91veO1BNTqGPvEIf7x2oBmDIwCgnJHs9zE5LJDoi1OVKA6vXhebuptAsbXyF8MaPYOdbED8KLv0JeD+l9c4iIiIiErSO1zexekcFuYU+8orKqTjRgDEwZXgcCzKTyfZ68A6OxfTi/+btlWuaRVzh8cKNf4cdbznh+bkbIXUuLHwAUia7XZ2IiIiISIfsqaght9BHbmEZ6/ZU0tRiGRAVxrwMD9neJOaN9TAoRrMqP0qhWaSj0i+G0fNh05OQtwQemw+TPw/Z/wkDhrhbm4iIiIjIRzQ2+1m/t7I1KPvYU1EDQLqnP7fMHUV2hoepqfG9uolXIGh6djs0PVs+Ud1RWPUwvPs7CAmDWd+AOXdCZKzblYmIiIhIH1Z+vIHlRU6n65XFFZxoaCYiLIRZoxPa1icPH9TP7TJ7BK1p7iSFZumQyj2Q82N4/0WISYL5P4Tzvwyh4W5XJiIiIiJ9gLWW9w8eI6fAR26Rj62lR7EWkgdEku111ibPGZNAvwhNMv4oheZOUmiWs1K6Ad74T9j3NiSkw8X/Dd7L1SxMRERERAKupqGZ1TsryG3dO9l33GniNWlYHAu8HrK8Hs5LGdCrm3gFghqBiXSnYdPg5qVQ9Bq89V/w3BdgxCy45CcwfLrb1YmIiIhIkCs5UtO2Nvnd3ZU0tviJjQzjorFJZHs9zMtIIrF/pNtl9koKzSKBYgx4F0P6pbD5Kch7EP54MYz7NGTfC4lj3K5QRERERIJEU4ufDXuryCvykVNQxq5yp4nX6KQYvjw7lSyvh+kjBxGuJl5dTqFZJNBCw2DaLTDhenj7Eeco+Bec/0WY9wMYkOJ2hSIiIiLSA1XWNLK8yEdOoY+VxeUcr28mPNQwc1QCX5iZSrbXw8jEGLfL7HO0prkdWtMsAXPCBysfgg1/gpBQmPkVmPMt6DfI7cpERERExEXWWgoOHSe3sIzcQh+b9ztNvJJiI8nKSCLbm8zc9ET6R2qssyuoEVgnKTRLwFXtdaZsb30OogY4wXnmVyFCLf9FRERE+oq6xhbW7Kwgt8hHXqGPQ9X1AEwaNpCs1i2hxqcMJCRETby6mkJzJyk0S5cpex9yfgLFr0H/wTDv+3D+l7RNlYiIiEgvtb+ylrwip4nX2l1HaGj2ExMRyoXpSWRnepifkYQnNsrtMvscheZOUmiWLleyFt76b9j/DsSPhHk/hInXO1O4RURERCRoNbf42bTvaGu36zKKy04AMDKhX9veyTNGDSIiTE283KTQ3EkKzdItrIUdb0Le/XBoCySOhfl3w7irIET/EhUREREJFlU1jawoLie30MeK4nKq65oICzHMGDWI7NZp16OT+rtdppxC+zSLBANjYOylkH4JFLwCeUvghZsh+WHI+hFkXObcIyIiIiI9irWWorLjzmhygY9N+6rwW0iIieCScc5o8tz0RAZEaQlesNNIczs00iyu8LfAthdh+RKo3A0p50P2PZCWrfAsIiIi4rL6phbe3lVBbqGPvMJyDhytA2D80AFkZ3jIzkxm4lA18QoWmp7dSQrN4qqWZtjyDKz4GVTvhxGzYP4PYdQ8hWcRERGRbnTwaF3r2mQfb++qoL7JT7+IUOaMSWSB10OW10PyADXxCkYKzZ2k0Cw9QnMDbHoKVv8Cjh1QeBYRERHpYi1+y+Z9VW1BufDwcQCGD4pmgTeZLK+HmaMGERWu5q3BTqG5kxSapUf5IDyv+jkcPwgjZreG54sUnkVEREQ6qbq2iRU7ysktKGNFcTlVtU2EhhimpcazINNp4pWW1B+j/+7qVRSaO0mhWXqkpnrY/BeFZxEREZFOsNay03eCnNbR5I0lVbT4LfH9wpmf4YTki8YmMTBaTbx6M4XmTlJolh6tLTw/DMcPOdO2L/qeGoaJiIiItKO+qYV3dh8hr9BHTqGP0iqniVfmkAFke5PI9iYzeXgcoWri1WcoNHeSQrMEhaZ6Z9r2mv911jwPneqE57GLFJ5FRESkzztcXd+2NnnNzgrqmlqICg9h7phEsrwesjI8pMRFu12muEShuZMUmiWoNDfClr8607aPlkDyBLjoO5B5JYSoSYWIiIj0DS1+y5bSo85ocoGP7YeOATA0LpoFmU6n61mjE9TESwCF5k5TaJag1NIE773gTNs+sgMSx8KF34Xx10BomNvViYiIiATcsfomVhVXkFNYxoqico7UNBJiYFrqILK8HhZkekj3qImXfJxCcycpNEtQ87fA9pdh5UPg2w7xI2H2nTD5CxCufQRFREQkeFlr2V1RQ26BM+16/d5Kmv2WgdHhzM9IItvrYd7YJOL6RbhdqvRwCs2dpNAsvYLfD0VLYfXP4cBGiPHArDtg2q0QNcDt6kREREQ6pKG5hXV7KtvWJ5ccqQUgIzm2bTR5yvA4wkJDXK5UgolCcycpNEuvYi3sWemE593LIXIgzPg3mPk16J/kdnUiIiIiH+M7Vk9ekROSV++ooKaxhciwEGanJZDtddYnD4vv53aZEsQUmjtJoVl6rQObYPUvoOAVCIuEKV+E2d+E+FS3KxMREZE+zO+3vHegum00+b0D1QAMGRjljCZ7PcxOSyQ6Qk28JDAUmjtJoVl6vYodzlZVW54D64fzroY5d8KQSW5XJiIiIn3EiYZmVu8oJ6fAR15RORUnGggxMGVEvDOanOEhc0ismnhJl1Bo7iSFZukzqg/AO7+BjX+GxuMwer7TNCwtW3s9i4iISMDtraghp9BHXqGPd/ccoanFMiAqjHkZHrK9Scwb62FQjJp4SdfrdaHZGHMR8F1gKpAC3GytffIT7h8J7DnNpcustcvO9HkKzdLn1B2FjU/Cu7+D44ecvZ5nfxPGfwZCw92uTkRERIJUY7OfDXsr24Ly7ooaANI9/cn2esj2epiaGq8mXtLtOhqag2nj1v7ANuCp1qOjFgFbTvm9MpBFifQa0XEw91twwdfgvb/B24/AS7dDzo+dc+d/SR23RUREpEPKjzewvMhHXpGPlcUVnGhoJiI0hAvSEvjSrFSyvcmMSFATLwkOQTPSfCpjzAngGx0caZ5urT3rIWONNEuf5/fDzjdhza+gZDVExDrBeeZX1DRMREREPsRay/sHj5FT4CO3yMfW0qNYC8kDIsnKcDpdzx2TSExkMI3ZSW/XG0eaz9WLxpgoYAfwC2vtC24XJBIUQkJg7ELnOLDJWfe87vfw7m8h80qY9XUYPsPtKkVERMQlNQ3NrNlZQW6hM6JcdqwBY2DSsDi+ffFYsrwezksZoCZeEvR6c2g+gbMGeg3QDFwJPGeM+bK19mlXKxMJNkPPh2seh4vvg3WPwcYnYPvLMGy6E569V0Bob/7XiYiIiADsO1JLbmEZuUXlvLPrCI0tfmIjw7hobBJZXg/zM5JI7B/pdpkiAdVrp2e389xvgLnW2ontXL8duB1gxIgRU0tKSjpbqkjv1HACtjzjjD5X7oaBI2DGbXD+FyE63u3qREREJECaWvxs2FtFXpGzd/JO3wkARifFsMDrTLuePnIQ4WriJUGo13XPPlUnQvOXgd9Za6PPdK/WNIt0gL8Fil+HtY86657D+8HEG2DmV8Hjdbs6EREROQeVNY0sL/KRU+hjZXE5x+ubCQ81XDA6gawMp9v1yMQYt8sU6TStaT69ycAht4sQ6TVCQsG72DkOvwfv/t4Zgd74hLPf88yvQvqlzn0iIiLSI1lr2X7oGHmFzmjy5v1OE6+k2EguGz+YbG8yc9MT6a8mXtJHBc3/840x/YExrb+GACOMMZOBSmvtPmPMg8AMa+2C1vu/DDQBmwE/cAXwdeAH3V68SF8weAJ8+tfOuudNT8K6x+GZz0L8SJhxO0z+grOtlYiIiLiurrGFNTsryCn0sbzIx6HqegAmDhvIXQvSyfZ6GJ8ykJAQNfESCZrp2caY+UDeaS792Vp7kzHmSWC+tXZk6/1fxgnIqUALUAz8b0ebgGl6tkgntTRB4b+c0ed9a1unbl8P02+DwePdrk5ERKTPKa2qJa/QmXb99q4jNDb7iYkI5cL0JLJbm3h5BkS5XaZIt+nVa5q7g0KzSAAdzIf1f4D3XoDmehh+gdM4LPNKCItwuzoREZFeqbnFz6Z9R8kt9JFbWEZxmdPEa2RCP7K9yWR7PUwfFU9kmJZRSd+k0NxJCs0dixpYAAAgAElEQVQiXaC2EvL/D9b/Ear2QIwHpn4Zpt4MA4e6XZ2IiEjQO1rbyIricnIKfKwoLqe6romwEMOMUYPI9jpNvEYn9Xe7TJEeQaG5kxSaRbqQ3w+7cmDdH2DHG2BCnGZiU2+G0VkQom0rREREOsJaS3HZCXIKy8gr9LGxpAq/hYSYCOZneFiQ6WFueiIDosLdLlWkx1H3bBHpuUJCIP0S56jaCxv+BJufhoJXnMZhU2+GKTdCTKLblYqIiPQ49U0trN11pDUol3PgaB0A56UM4BtZY8jyepg0LE5NvEQCRCPN7dBIs0g3a25wQvOGJ5w9n0MjnDXP026G1Dlg9Ae/iIj0XQeP1pFb6COv0MeaXRXUN/mJDg9lbnoi2V4PWRkeBg9UEy+Rs6GRZhEJLmGRMOFa5ygvcsLzlr/Cthcgcawz+jzps9BvkNuVioiIdLkWvyV/fxW5hT5yCnwUHj4OwPBB0dwwbTjZmcnMHDWIqHA18RLpahppbodGmkV6gMZa2P6y0zjswAYIjYTMK5zmYalztfZZRER6leq6JlYWl5PbundyVW0ToSGGaanxbU28xnj6YzT7SiQgNNIsIsEvoh9M/rxzHN4Gm56Crc86o8/xo+D8LznXYge7XamIiMhZs9ay03eidUsoHxtKqmjxW+L7hZOV4SHL6+Gi9CQG9lMTLxE3aaS5HRppFumhmuqctc8b/+ysfTahkHGZE6DTFkCo/i5QRER6rvqmFt7dU0luQRm5RT72VzpNvDKHDCDbm0S2N5nJw+MIVRMvkS6nLac6SaFZJAhU7ITNT0H+X6GmHGKHOOueJ98IiWPcrk5ERASAw9X15BU5a5PX7KygrqmFqPAQ5o5JJKu1iVdKXLTbZYr0OQrNnaTQLBJEmhuheBnk/5+z77P1w4hZMPkLcN5VEBnrdoUiItKH+P2WLaVH26Zdv3/wGABD46KdtcmZHmaNTlATLxGXKTR3kkKzSJA6fhi2POvs+3xkB4THwHlXw5QvOEFazVNERKQLHKtvYlVxRVsTryM1jYQYmJoaT7Y3mWyvh7HJauIl0pMoNHeSQrNIkLMW9q+D/Kdh24vQeMJpHjbpc84U7vhUtysUEZEgZq1ld0UNuQXOaPL6vZU0+y0Do8OZn5FEttfDvLFJxPWLcLtUEWmHQnMnKTSL9CKNNbD9n86+z3tWAdbZsmrSZ2HcpyFqgNsViohIEGhobmHdnsq2adclR2oByEiOJcvrYUGmhynD4wgL1ZaIIsFAobmTFJpFeqmj+2Hrc7DlGTiyE8Kinb2fJ30WRs+HEK0vExGRk3zH61le6OydvGpHOTWNLUSGhTA7LYFsr7Mt1LD4fm6XKSLnoMtCszEmEkgBooFya235uZXYsyk0i/Ry1sKBjU7n7W1/h/qj0H8wTLgWJt4Agydo/bOISB/k91u2Hawmp8BHXpGPraXVAAwZGOWMJns9zE5LJDpCf8kqEuwCGpqNMbHAjcDngBlAOGAACxwAXgces9au70zRPYlCs0gf0twARa/Be3+D4tfB3wRJXph4PUy4DuJGuF2hiIh0oRMNzazeUd4alMupONGAMTBleBwLMpPJyvCQOSRWTbxEepmAhWZjzLeBHwG7gX8C64CDQB0wCBgPXAhcDbwDfNNau6NT1fcACs0ifVRtJWx/GbY+D/vWOudS5zjh+byrIDre3fpERCQg9lbUkFPoI6/Qx7t7jtDUYomNCmPe2JNNvBL6R7pdpoh0oUCG5ueBH1trt53hvkjgVqDRWvv42RTbEyk0iwhVe53R563PQ0UxhITDmIudKdwZl0FEjNsViohIBzU2+9mw92QTr90VNQCM8fR31iZneJg2Mp5wNfES6TO6ZE2zMSapt65h/iiFZhFpYy0cyof3XnC2rzp+EML7QcZiJ0CnLYAwbSkiItLTVJxoYHlRObmFZawqruB4QzMRoSFckJZAdkYS2d5kRiSoiZdIX9VVoXkHsNBau7szxQUDhWYROS2/H/a97QTo7S9DXRVExcG4K2H8tTByrjpwi4i4xFrL+wePtY0mbyk9irWQPCCSrAwP2V4Pc8YkEhMZ5napItIDdFVo/iVwA7DYWrvplPMXAQ9aa+ecS7E9kUKziJxRcyPsznMCdOGr0FQDMR4nQJ/3GRgxC0I0zU9EpCvVNjazekcFeUVOUC475jTxmjQsjmyvE5TPSxmgJl4i8jEdDc1n9dds1tq7jDH7gVxjzPWAD/gpcAnw/DlVKiISrMIiYOxC52ishR2vw/svweb/g/WPO1tYnXeVE6CHTVeAFhEJkH1HasktLCO3qJx3dh+hsdlPbGQYF45NJNubzPyMJBLVxEtEAuSs56ZYax8yxoQC/8LZduplYKK19v1AFyciEjQi+sF5VztHwwkoXuYE6A1PwLu/gwHDnAA97tMwdJoCtIjIWWhq8bOxpKpt2vVO3wkARifG8KULUsn2epg2chARYfp3q4gE3tlOzx4O3APcBGwAJgHfsNY+2RXFuUnTs0UkIOqPOQF624uwKwdaGiE2xZnCPe7TMHym1kCLiJxGZU0jy1unXK8oLud4fTPhoYaZoxLapl2PTNQuBiJy7rpkejawA9gKfMpa+6YxJht40Rgz1Fr7wLkUKiLSq0UNgInXO0d9NRS/Dtv/cXIEun8yZF7hBOgRsyFUzWlEpG+y1lJw6Dh5RT5yCsrYvN9p4pUUG8ll4weT7U1mbnoi/dXES0S62dmONF9jrf37R85NApYC/7DW3hHg+lyjkWYR6VINx2HHG06ALn4DmuugX4KzjVXmlTB6HoRpPZ6I9G51jS28vauCnEIfeYU+DlXXAzBx2MC20eTxKQMJCVETLxEJvC7pnv0JH5YKvGatHdfpl/UQCs0i0m0aa2DnWycDdONxiIiFsZeC91OQfglExrpdpYhIQJRW1ZJX6COn0MfaXUdoaPYTExHKhelJZHs9zPcm4YmNcrtMEekDujU0t35gvLW2KiAv6wEUmkXEFc0NsHsFFL4ChUuhtgJCIyEtywnQGZdBTKLbVYqIdFhzi59N+46S2zqaXFR2HIDUhH5kez0s8CYzfVQ8kWHq7yAi3StgodkYM8pau6eDH2qAYdba/R0rs+dSaBYR1/lbYN87UPAKFP4LqveDCXGah2UsBu/lkJDmdpUiIh9ztLaRFcXl5BQ4Tbyq65oICzFMHzmIBZkesrweRifGaO9kEXFVIEPzYeBV4HFr7dp27okHPgvcCTxqrf312Zd8ZsaYi4DvAlOBFODmM3XuNsZMAH4NzAAqgd8DP7Fn+OIKzSLSo1gLh7ZA0VJnBLrsPed8YgZ4F0PG5TB0qrayEhFXWGspLjtBTmEZeYU+NpZU4beQEBPB/AxnbfKFYxMZEBXudqkiIm0C2T3bC/wIeNUY4wc2AgeBeiAeGAdkAuuAb1lrXz/nqs+sP7ANeKr1+ETGmAHAm8BKYDrOd3kCqAEe7royRUQCzBhImewcWf8BVSVQ9BoUvQprfgWrf+F04h67EMZeBqPnO3tHi4h0kfqmFtbuOtK2d/KBo3UAnJcygG9kjSHL62HSsDg18RKRoNfhNc3GmGjgcmAukApEAxXAZuB1a+22riqynXpOcIY9oo0xXwN+BiRba+taz90DfA1nGnm7X14jzSISNOqqYMebUPgq7MxxGomFRcGoea0hehEMHOp2lSLSCxyqrnNCcoGPNbsqqG/y0y8ilDljEsn2esjK8DB4oJp4iUhw6PZGYN2tg6H5KSDBWnv5Keem44yKj/6ktdoKzSISlJobYd/bULQMil+Dqr3O+cETnBHosYsgZYqmcYtIh7T4Lfn7j5JbWEZuYTkFh44BMHxQNAu8yWR5PcwcNYiocDXxEpHgo9Ds3PMGUGqtveWUcyOAEmD2R9doG2NuB24HGDFixNSSkpKuKF1EpHtYCxXFzjTu4mWw/12wfuiX6GxjlX4ppGVDdJzblYpID1Jd18TK4nJyC50mXpU1jYSGGKalxjvdrjM9pCX1VxMvEQl6gVzTfOpL5wE/B5KAnUD+Kcd2a23zOdTaY1hrHwMeA2ek2eVyREQ6xxhIynCOud+C2kpnP+gdbzghesszYEJhxAUnQ7RnnPOciPQZ1lp2+k60rU3eUFJFi98S3y+c+RlOp+t56UkM7KcmXiLSN51VaAb+iNOI6xfAGGAKcC0wDGgEetoilsNA8kfOJZ9yTUSk7+g3CCZe7xz+Fijd4AToHa/DW//tHAOGwZgFMOZiGD0Poga6XbWIdIH6phbe3VNJbkEZuUU+9lc6Tby8g2P56rzRZHs9TB4eT6iaeImInHVoHgxcaq3dfepJY8wgnADd06wFfmaMibLW1reeuwSn+/de16oSEXFbSCiMmOkcC/4Tjh08OQr9/kuw6c/OKPTwmSdD9OCJWgstEsTKjtW3jSav3lFBXVMLUeEhzE5L5CsXpZHl9TA0LtrtMkVEepyzWtNsjHkV+Lm1NqfrSvrEz++PM8IN8DbwU+CfQKW1dp8x5kFghrV2Qev9A4EiYDlwPzAWeBK4z1r7iVtOqRGYiPRZLU1Qut4J0TvfcvaHBohJgrQFzjrotCzo73G3ThH5RH6/ZUvpUfIKfeQU+nj/oNPEa2hcNNleZ+/kWWkJauIlIn1WwBqBtTbTyge2AE3AHcB11tryQBR6Nowx84G801z6s7X2JmPMk8B8a+3IU56ZADwKzACqgN8BP/6k7aZAoVlEpM0JH+zKbQ3ROVBX6ZxPnuCE57RsGDELwnvaCh2RvudYfROrd1SQU+BjRbGPihONhBiYmhpPltfDAm8yY5PVxEtEBAIbmh8EJrceH6wHPgq8gjP9eTOw9YN9kHsLhWYRkdPw++HwFidE78qDfe+Av8nZFzp1zslRaDUUE+kW1lp2V9Q4o8kFPtbvraTZbxkYHc78jCSyvR4uSk8iPibC7VJFRHqcLtlyyhiTzMkA/cGRDlhgh7V23LmV2/MoNIuIdEDDCShZ0xqic50trgBiPE4jsdHzYdQ8iBvuZpUivUpDcwvr9lSSW+gjr9DH3iO1AGQkxzqjyZkepgyPIyxUPQhERD5Jt+3TbIyJBiYBk6y1v+/Uy3oQhWYRkXNwdD/sWQG7l8PuFVDjc84PSnMC9Oj5MOpCiI53rUSRYOQ7Xs/ywnJyCstYvaOCmsYWIsJCmJ2W0LY+eVh8P7fLFBEJKt0WmnsrhWYRkU6yFnwFrQF6OexdDU01gIEhE2HURc4o9IhZENnf5WJFeha/37LtYDU5BT7yinxsLa0GYPCAKLIzPWRneJg9JoF+EWe7EYqIiHxAobmTFJpFRAKsuREObIA9q2DPSihdBy2NEBIGQ6e2huiLYNgMNRWTPulEQzOrd5Q7066Lyik/3oAxMGV4HAsyk8nK8JA5JFZNvEREAkShuZMUmkVEulhjLex/1wnQe1bCwc1gWyA0EoZNh5FznWPYdIVo6bX2VtS07Z387p4jNLVYYqPCmDfWaeI1b2wSCf0j3S5TRKRXUmjuJIVmEZFuVn8M9q11AvTe1XB4K1j/KSF6zikhOtrtakXOSWOznw17nSZeuUU+dpfXADDG079tbfLU1HjC1cRLRKTLdXtoNsaMAEqttf6AvNBlCs0iIi6rr3a2tNq7ygnRh7a0hugIZzp36mznGD4TImPdrlakXRUnGsgrdNYmryqu4HhDMxGhIcwcPYjs1r2TRySoiZeISHdzIzT7gfeBO6y1qwLyUhcpNIuI9DCnhuiStSenc5tQp7FY6hwnRI+YBf0GuV2t9GHWWt4/eIzcQh85hT62lh7FWvDERraNJs8Zk0hMpJp4iYi4yY3QfBMwClhkrZ0ZkJe6SKFZRKSHazgBpeuh5G3nKF0PLQ3OtaRMGHGBE6BHXABxI0DNk6QL1TY2s3pHRWsTLx9lx5wmXhOHxbGgNSiflzJATbxERHoQrWnuJIVmEZEg09wABzZByRpnRHr/u9BwzLkWm/LhEJ18HoSEuluvBL39lbVto8nv7D5CY7Of2MgwLhybSFaGh/kZHpJi1cRLRKSn6mho1rwgERHpHcIiIXWWcwD4W5x9ovetdY6StfD+i861iFgYNhWGXwDDZ8CwaRA10L3aJSg0t/jZWFLV1u16h+8EAKMTY/jSBalkez1MGzmIiDA18RIR6U3OeqTZGBMJpADRQLm1trwrCnObRppFRHoZa6F6vxOe978L+9dB2TbAAsYZfR4+w2ksNnwGxI/SlG6hsqaRFcU+cgvLWVHk41h9M+GhhpmjEshqnXY9KjHG7TJFROQcBHSk2RgTC9wIfA6YAYQDBrDGmAPA68Bj1tr1516yiIhIFzLGWdscNwIm3eCcqz8GBzY4AXr/u7D1b7DhT861fonO9lbDpzs/U86HyP7u1S/dwlpL4eHjbaPJm/dV4beQ2D+SReMHk+31MDc9if5q4iUi0meccaTZGPNt4EfAbuCfwDrgIFAHDALGAxcCVwPvAN+01u7owpq7hUaaRUT6IH8L+LY7TcX2r3d+Hmn9I82EOKPRw6bDsNYp3YPSIERTcYNdXWMLb++qIKfQR16hj0PV9QBMHDaQrAwPCzI9jE8ZSEiIZh6IiPQmAWsEZox5HvixtXbbGe6LBG4FGq21j59NsT2RQrOIiABQWwkHNjqj0aXrnf/9QYOxqIHOCPSwaTB0mvMzJtHdeqVDSqtqyWsdTX571xEamv3ERIQyNz2RBd5k5mck4RkQ5XaZIiLShbqke7YxJqm3rmH+KIVmERE5LX8LVBRD6QZnaveBjVC23dkzGiAu1QnPKefD0PNhyCSI0JpXtzW3+Nm8/yg5Bc5oclHZcQBSE/qR7fWwwJvM9FHxRIapq7qISF/RVaF5B7DQWru7M8UFA4VmERHpsMYaOLTlZJAu3QjHSp1rJsTZN3rolJNB2nMehEW4W3MfcLS2kRXF5eQW+lheVE51XRNhIYbpIwexINNDltfD6MQY7Z0sItJHddWWU0uBt40xi621m075sIuAB621c87yfSIiIsEvIgZSZzvHB074nH2jD25yfhYuhc1PO9dCI2HweEiZAkMmOz+TvBCq5lKdYa2luOxEaxOvMjaWOE28EmIiuDgzmWyvhwvHJjIgKtztUkVEJIic1Z/O1tq7jDH7gVxjzPWAD/gpcAnwfBfUJyIiEpz6eyBjkXOAs+XV0RInQB/Y6IxMb3kO1re2AQmLgsETTgnSkyExQ0H6DOqbWli760hbt+sDR+sAOC9lAF/PGkO218OkYXFq4iUiIufsrP8kttY+ZIwJBf6Fs+3Uy8BEa+37gS5ORESk1zAG4kc6x/jPOOf8fqjcDQc3O8ehfMj/K6x7zLkeFgXJ45110R8cnkwIi3TrW/QIh6rrnJBc4GPNrgrqm/xEh4cyZ0wi38geQ1aGh8ED1cRLREQC46xCszFmOHAPcBOwHpgEvKrALCIicg5CQiBxjHNMvM4552+BIzudkegPjvf+Bhv+2PpMuBOch0yCwRNhyEQnWPfiPaRb/Jb8/UfJLSwjt7CcgkNO9/Jh8dHcMG042ZnJzBw1iKhwNfESEZHAO9uR5h3AVuBT1to3jTHZwIvGmKHW2gcCX56IiEgfExIKSRnOMfF655zfD1V7TgnS+VD4Kmz+S+tDBhLSnOndgyeeDNP9Pa59jc6qrmtiZXE5eYU+lheXU1nTSGiIYWpqPHdf5iXb62GMp7+aeImISJc729D8BWvt3z/4xVqba4yZByxtDc53BLY8ERERISTECcUJaSendlsLxw7C4a1w+D0nTB/YBO+/dPK5GI/TcCx5vBOok8dDYjqE9rxGWNZadpU7TbxyCnxsKKmixW+J7xfO/Ayn0/W89CQG9ut5tYuISO92VltOtfsSY1KB16y14zpfUs+gLadERCQo1R11QvTh96Bsm/OzvBBaGp3roZHOKPYHITr5POeISez2UhuaW3h3d2VbE699lbUAeAfHsiDTQ7bXw+Th8YSqiZeIiHSBLtmn+QwfGG+trQrIy3oAhWYREek1WpqgYkdrkH4PDm9zAnVN+cl7+iefDNCe1p9JGQFvOlZ2rJ681pC8emcFtY0tRIWHMCctkSyvE5RT4qID+pkiIiKnE7B9mo0xo6y1e850n7W2yjgLi4ZZa/d3sE4RERHpaqHhkDzOObjh5PkTPih73zl8250g/e5j0NLgXDehzpRwz7jWI9MJ0/EjnbXXHeD3W7aUHnWCcpGPbQecJl5D46L5zPlDWeBNZlZagpp4iYhIj9WRNc1rjTGvAo9ba9ee7gZjTDzwWeBO4FHg14ErUURERLpEf49zpGWdPNfSDJW7TgZpX4GzXnr7P4DW2Wlh0c4otCcTkrwnfw4cDiEhHKtvYvWOCnIKfKwo9lFxopEQA+ePiOf7izLI9nrISI5VEy8REQkKHQnNXuBHwKvGGD+wETgI1APxwDggE1gHfMta+3oX1SoiIiJdLTTsZPduPnPyfGONszbaVwBl28H3PuzKgy3PtN3SFBpNSchw8uuHUOQfSl14KpePnsSU8ROZl5FMfExE938fERGRTurwmmZjTDSwGLgQSAWigQpgM/C6tXZbVxXpBq1pFhERaV9js591eyp5e9tO9hdtJPb4LtJNKZMjD5EeUkr/psqTN4fHQNJYZzQ6KcP5mTj2rKZ5i4iIBFrA1jR/wFpbZ4wpB+611h7rVHWdYIy5A/geMAR4H2d0e1U7984H8k5zKdNaW9hlRYqIiPRCvuP1LC8sb2vidaKhmYiwEGanTWf6vMvJyvAwfFA/5+aaI1BR5IxOlxc5x+4VHxqZJjQSEsY4gTrx1CMdwtUMTEREeoaz3af5LSDUGFMCbAHyP/hprd0b4No+xhhzA/BL4A5gdevP14wx46y1+z7h0fOAU/7Km/L2bhQRERGH32/ZdrC6bUuoraXVAAweEMUVk1JY4PUwe0wC/SJO858TMQkQMxtSZ3/4fH01lBc7YbqiyOnqfTDfWTNt/a03GYgbDokZToBOGHMyTPdPBq2FFhGRbnRWW04ZY2YBz+GM3jbgrGWeg9MZpBz4G/CAtfZw4EsFY8y7wFZr7W2nnNsBvGCtvfs0989vrTXJWltxNp+l6dkiItIXnWhoZvUOZzQ5r6ic8uMNGANThseR7fWQ7U0mc0gXNPFqqncakFUUO6G6otgJ1Ud2QVPtyfsiB7SG6HRISIfEMc7vg0ZDRExgaxIRkV4t4NOzW/0e+Kq1dukpHzQV+Avw5P9v786j4zrPO89/X+zESoJAgSu4k0XJtiSL1r4BNGNbPp5x7MRLxoucjtW2YneU2E57m4k7456cJG4vk3Qfj2d6TvZuHSeZJZ147JgAtVsSZdmKbAKkSIqUuKAAggsWYn/nj1tYCBPcRKBQ4Pdzzj236q33Fh6KV8X64d77XKAZeCGEcHOM8ehlvvcFhRBKgJuBr0176QfAHb+4xTl2hxBKgZ8DX40xnu+UbUmSrkmvdPVNHE1+5uAJhkcjVWVF3LO5nu3pFPdurmdp5dW9X/MvKC6bvE/0VGNj0HM0G6JfTtYn9sErT8KLj5w7t3plEqDPWTbA4sbktluSJF2Byw3NG4H9UwdijM+HEH4HeCjG+PYQwt8AfwB89CrVOK4OKAQ6po13AG+dYZtjwCeB54AS4MPAzhDCvee7DjqE8CDwIEBjY+NVKluSpPlleHSM517ppmVPcu/kA519AGxMVfKxO9fRnE5x85olFBcW5LhSoKAAalYly4bmc18b6kuORJ94ecr6ZXjpb5PTwCfeoygJzrUbkhBduwGWrk/WixttRiZJuqDLDc3PAg8A00+F3guM3+TxG8Dfvr6yro4YYzvQPmXo6RDCWpJGYr8QmmOM3wG+A8np2XNQoiRJc6Krd5Bd7Z20tmV4bG8nPYMjlBQWcNuGpXzktjU0pxtoXFqe6zIvT0kFLH9TskwVI/SfmAzT3fsn14eeguG+ybkFxbBkTXJ69/TFI9SSJC4/NH8GaAkhbAB+P8b4UvZWVJ9hsrlWF1B/FWsc1wWMAg3TxhuAy7mG+hngA1erKEmS5qMYIz87eobWtgw72zL89LVTxAipqlLe+ablNKVT3LWxjorSy/0qkAdCgIq6ZGm87dzXYoTejskQ3X1gcjn0FAz1TnmfwuQId+06WLJucr1kbfK4tGpO/1iSpNy4rH8ps6di3wr8r8CLIYRhklOmh4CPZKe9Gbiq1zNnf/ZQCOF5YAdJw7FxO4C/u4y3upHktG1JkhaU/qERntjXRWt7hta2To6fGSAEeNOqxfz2WzfTnE5x/Yrqq9/EK5+EAFXLkmXtnee+FiP0dWZD9MHJMH3yYNLd+2z3ufPL67JBei0sXpOsx5fqFZ72LUkLxGX/ejl7f+NfCiGsIgnIo8DuGOP4tcbHgX979Uo8x9eBvwwhPAs8CXwCWAF8GyCE8BfZGj+Sff4w8ArJ/ZxLgA8B7wbeO0v1SZI0pw6f6KelrYOW9k5+dOAEQyNjVJYWcfemOprTKe7bkqK+apabeC0UIUBlKlmmH6GG5Drp7oNJiJ66fvUZeOnvIY5Ozi0oTm6bNRGo1ySney9emzwuX+qtsyQpT1zxOVkxxteA184z/uTrqujCP/OREMJS4MvAcuAl4P4Y46HslOndu0qAPwZWAWdJwvM7p3b/liQpnwyPjvH8oZMTp12/nElOJ15fV8GHb1vD9nSKbWtrKSmaB028FpqyGlhxY7JMNzoMp1+Dk68ky6lDk4+P/uQXj1IXV2SD9HiYnrYsWmKolqR54rLu03wt8T7NkqT5ortviEf3Zti5J2nidWZghOLCwC3ramlON9CcTrGuznsUz2uDPXDyEJw6nA3Uh6asD8NQz7nzSyonA3TN6uSodc3qyecV9UlncUnSFZut+zRLkqRZFmOk7XjPxL2TXzh8krEIdZUlvO36ZTSnU9y1qY6qMjs7543SKlj2hpvAsygAACAASURBVGSZLkYYOJUN1OPLq5MB+9DTMHj63G0KS6Fm5bmBevzWXDWrk3tWF5fNzZ9NkhY4Q7MkSfPA2aFRntrfRUtbhta2DEdPDwDwxpU1fLp5E83pFG9cWUNBgafsLjghJKdjL1oCy284/5yB08np36dehdPZQH361eT5vn9OOoJPV1GfhOfxMF29Mgna1auSdeUyKPSroCRdjJ+UkiTlyJFTZ5OjyXs6eGr/CQZHxqgoKeSuTXU8/NbN3LelnlS1RwtFcj11WQ00XH/+10cG4czRJFhPLK8m6659cGDXubfTguSWWlXLpoTplUnX7+oVSbCuXgGVDQZrSdc8PwUlSZojo2ORFw6fZGf2aHLb8eQ61jVLy/m1WxtpTqe4ZV0tpUXeqkiXqag0uf1V7brzvx5jcrT6zBE4fQTOvJZdH0mC9bEXof3/g5Gz524XCpIj0tUroHo5VK2YDNZVyycfFy+a/T+jJOWIoVmSpFl0qn+IR/d20tKW4dG9nZzqH6aoIPCWtbV86f6tNG9Nsb6u4tq+d7JmXwiwaHGyzHS0OkY4ezI5Yn3maBKszxxNwnXPUejcCwcehcEzv7ht2eJsiM4G66pl2cfLJ8N1Rb33rpaUlwzNkiRdRTFG9nb0TlybvPtQN2MRaitKaE6n2J5u4O7NdVTbxEvzTQhQXpss52tYNm6wB84cS4L0mWPJ0eqeY9BzPAnZmTboPQ5xbNr7F0BFKgnUVcuhqiG7XpYcza5qSNYV9Z4SLmle8RNJkqTXaWB4lKcPnKBlT9Lt+sip5BTX65ZX85tNG2lKp7hh1WIKbeKlhaC0CuqroH7zzHPGRqE3kw3Tx5Iw3duRBOue48kp4Ud2Q1/neTYOSXCuakiuqZ4aqCtTydj4ayXeak3S7DM0S5J0BY6dPjtxNPmJl7sYGB5jUXEhd25cykNNG2hOp1he43WeukYVFCanZ1cvv/C8kSHoy0yG6d7j0NNx7vr4S8mc6UeuIbmfdeV4uK5P1hWpbLjOLuPPi0pn588qacEzNEuSdAlGxyI/efUUrW0ZdrZl2HMsua5z1ZJFvG/baprTKW5bv5SyYq/ZlC5ZUcnkLbEuZGwU+ruTEN3bkRzFnrru6YDMnqRL+MDp879HaU0SrCtSk+uK+iljKaioS8ZKKpPT1SUJQ7MkSTM6fXaYx/Z20tqWYdfeTrr7higsCNzcuITPvyPN9nSKjalKm3hJs62gMHskuR5444XnDg8kp333ZaaE6ynP+zqh4+fQt2vmgF20KAnP4yH6nMd1UF6XfZ59XOyt4aSFzNAsSVJWjJH9nb3szF6bvPvQSUbHIovLi2nakqIpneLeTfXUlNvES5q3istg8epkuZiRoSkBuxP6u7LPO6Ev+7j3OBz/l+Tx2PD536ekCiqWTobp8rrs8xnGPJIt5RVDsyTpmjYwPMozB7uzp1138Gp30sQrvayKf33PerZvTXHj6iU28ZIWoqISqFmZLBcTY3K7rb6uyUDd3zXt+Ymkm/ixF5PXRofO/16FJZOBurw2+3jqku1ivqh28nlxuUFbyhFDsyTpmtNxZmDi2uQnX+6if2iU0qIC7txYx4P3JE28Vi62iZekKUKAsppkWbrh4vNjTG7P1d+VXI/d15WE6oklO95/Ao6/mLw+cGrm9ysqy4boWli0ZEqoPt96SfK4rMbbd0lXgf8XSZIWvLGxyItHTtOyp4OW9gwvHUmaeK1cvIj3vHklzekUt6+vY1GJTbwkXSUhQFl1stSuv7RtRkeS4DwepvtPwNnxx93Jcja7zuzJPj8JcXTm9yyryYbobJCeeHy+ZXGyLlucHIWXBBiaJUkLVM/AMI/v66KlLcOu9gxdvUMUBHhz4xJ+9+1baE6n2NJQZRMvSfNHYdFkg7FLNTaWnDZ+thv6TyYh+mw2TI+H6vGx/m44eTD7/BQQZ37f4vLJAD01TC9aPGVdc54xA7cWHkOzJGnBONDZm9w7uT3Dswe7GR6NVJcVcd+WFNu3prhnUz1LKvwyJ2kBKSjIhtrFUHsZ242NweDpKaE6G6TH1wOnpqxPQnc2bA+cguH+C7930aLsEe4pwXr81PZFi6G0evL5Ocvi5Mh8oc0WNb8YmiVJeWtoZIxnD3ZPBOWDXX0AbG6o5NfvWsf2dANvblxMUWFBjiuVpHmmoGDytOzLNTKU3K5rarAeOD0ZqgdOZ59nH/d2QFf75Hgcu/D7F5dng3U2XJ/vcWlNdl09uS6tmpzjtdy6itybJEl5JdMzwK72Tlr2ZHji5S56B0coKSrg9vVL+dida2nakmJ1bXmuy5SkhauoZMp9sy9TjDDUOxmgJ5Yzk0F84HRyyvnUsVOHJ8dHBi7+c4rLkxA9Eaaz69Ka7Hr6Uj05p6Ryctyj3sLQLEma58bGIj87eoadbR20tmX46WunAWioLuVdN6ygOZ3izo1LKS/xnzRJmvdCmAykNauu7D1GBpPO5FPD9dT1YM+U5z2TYz0d5z6/0DXd44rKzg3R40tJJZRWZtfVUx5Pf70iuY93aWXyXvbRyEt+w5AkzTu9gyM8sa+L1rYMLe0ZOnsGCQFuXL2Yz+zYTPPWFNctr7aJlyRdi4pKk+VyGqZNNzYGw33ZEN1zbpge7IHB3smxod5zx88cSdZDvcl65Oyl/cxQOBmup4fqkopkmfp6ScWU9fTH2ec2XZsThmZJ0rxw6EQfO/ck1yY/c6CbodExqkqLuGdLPc1bUty3pZ6llaW5LlOStBAUFEweFX69RkeSAD0RrnthqAeG+qaE657snL5zXx/qS049H99+qO/ijdbO+XMUQ0n5uYG6eDxUl5/7eOK18inr8fHyaeMVyX8jAYZmSVKODI+O8dwr3bS2ZdjZluFAZ9LEa0N9BQ9kr03etnYJxTbxkiTNZ4VFkx3Mr4ax0WyA7s8G697JgD3xeOpYX3LUfOJ5P/QeT9bD/ZPjY8OXV0dR2ZRQvejcgF286Nzx844tgk2/tCCuCzc0S5LmTFfvILvaO2lty/DY3k56BkcoKSzg1vW1fOS2NTSnG2hcahMvSdI1rKBw8jZcV9Po8JSQ3T9l3Z+E8Oljw30wfHYyfI+P95+YMqc/mTPcz3mvEf/iUUOzJEkXEmPSxGv8aPJPXztFjJCqKuWdb1pOUzrFXRvrqCj1nyNJkmZVYfHVPSI+VYxJV/Phs9ngnQ3SxQvjF+F+S5EkXVX9QyM8+fIJWto6aG3r5PiZ5NYgN6xezMPbN9OcTnH9imoKCmziJUnSghBC9rTsRVBem+tqrjpDsyTpdXu1u5+WtgwtbRmePnCCoZExKkuLuGdzHU1bUty3JUV9lU28JElS/jE0S5Iu28joGM8fOklLe4aWPRn2ZXoBWF9XwYdvW8P2dIpta2spKbKJlyRJym+GZknSJTnZN8SjezvZ2Zbh0fYMZwZGKC4M3LKulg/c0khzOsW6uopclylJknRVGZolSecVY6TteM/EadcvHD7JWIS6yhLedv0ymtMp7tpUR1VZ/nfFlCRJmkneheYQwkPA54DlwM+Ah2OMj19g/r3A14HrgaPAH8UYvz0XtUpSvjk7NMpT+7toacvQ2pbh6OmkidcbV9bwqeZNNKdTvGlljU28JEnSNSOvQnMI4f3At4CHgCey6++FEK6LMR4+z/x1wD8B/yfwIeAu4D+FEDpjjH83d5VL0vx15NTZ5Gjyng6e2n+CwZExyksKuWtjHb/11k00bUmRqi7LdZmSJEk5kVehGfgd4M9ijP979vmnQwhvBz4JfOE88z8BHI0xfjr7fE8I4Vbgs4ChWdI1aXQs8sLhk+zMHk1uO94DQGNtOR/MXpt86/paSosKc1ypJElS7uVNaA4hlAA3A1+b9tIPgDtm2Oz27OtTfR/4aAihOMY4fHWrlKT56VR/0sSrtS3Drr2dnOofpqggsG3tEr50/1aa0ik21FcQgqddS5IkTZU3oRmoAwqBjmnjHcBbZ9hmGfDD88wvyr7fsakvhBAeBB4EaGxsfJ3lSlLuxBjZl+nNnnad4fnDJxkdi9RWlNCcTtGcTnH3pnpqFtnES5Ik6ULyKTTPuhjjd4DvAGzbti3muBxJuiwDw6P86MCJiW7Xr508C8D1K6p56L4NSROvVYsptImXJEnSJcun0NwFjAIN08YbgOMzbHN8hvkj2feTpLx2/PRANiR38OTLJzg7PMqi4kLu3FjHQ/dtpCldz/KaRbkuU5IkKW/lTWiOMQ6FEJ4HdgDfnfLSDmZu6vU08MvTxnYAu72eWVI+Gh2L/OTVU7Rmjyb//NgZAFYtWcSvbltFUzrF7euXUlZsEy9JkqSrIW9Cc9bXgb8MITwLPEnSHXsF8G2AEMJfAMQYP5Kd/23gUyGEbwL/G3An8ADwwbktW5Ku3Omzwzy+r5OWPUkTr+6+IQoLAjc3LuHz70jTnE6xKVVpEy9JkqRZkFehOcb4SAhhKfBlYDnwEnB/jPFQdkrjtPkHQwj3A98guS3VUeDfeI9mSfNZjJH9nX20tHXQ0pbhuVeSJl6Ly4u5d3M9zekU926uZ3F5Sa5LlSRJWvBCjPa7Op9t27bF3bt357oMSdeIwZFRnjnQPdHE63B3PwDpZVU0pVNsT6e4cfViigoLclypJEnSwhBCeD7GuO1i8/LqSLMkLSQdZwYmrk1+4uUu+odGKS0q4M6NdXz8nvU0p1OsXGwTL0mSpFwyNEvSHBkbi7x45DQtezpoac/w0pGkideKmjLe8+aVNKdT3L6+jkUlNvGSJEmaLwzNkjSLegaGeXxfFy1tGXa1Z+jqHaIgwJsbl/C5t21h+9YUWxqqbOIlSZI0TxmaJekqO9DZO3Ft8rMHuxkZi1SXFXHvluTa5Hs317OkwiZekiRJ+cDQLEmv09DIGM+9MtnE62BXHwCbGyr5V3evo3lLipvXLLGJlyRJUh4yNEvSFejsGaS1PUNrW4bH93XROzhCSVEBd2xYysfuXEvTlhSra8tzXaYkSZJeJ0OzJF2CsbHIz46eyR5N7uCnr50GYFl1Ge+6YQXb0ynu2LiU8hI/ViVJkhYSv91J0gx6B0d4Yl8XrW0ZWtszZHoGCQFuXL2Yz/7SZprSKa5bXm0TL0mSpAXM0CxJUxw60TdxbfIzB7oZGh2jqrSIe7bU07wlxX1b6llaWZrrMiVJkjRHDM2SrmnDo0kTr9a2DDvbMhzoTJp4baiv4KN3rKE53cC2tUsotomXJEnSNcnQLOmac6J3kF3tnbS0ZXhsbyc9gyMUFwZuXbeUD9+2huZ0ijVLK3JdpiRJkuYBQ7OkBS/GyM+PnaFlT4aW9gw/efUUMUJ9VSn3v3E5TekUd22qo7LUj0RJkiSdy2+Ikhak/qERnnz5BC1tyW2hjp8ZAOCGVTU8vH0zzekU16+opqDAJl6SJEmamaFZ0oLxanc/re0Zdu7J8PSBEwyNjFFZWsTdm+poSidNvFJVZbkuU5IkSXnE0Cwpb42MjvHjw6fY2dZBa1uGvR29AKyrq5i4Nvkta2spKbKJlyRJkq6MoVlSXjnZN8Sje5MmXo/u7eT02WGKCgK3rKvlfdtW05xOsb6+MtdlSpIkaYEwNEua12KMtB3vmbg2+ceHTzIWoa6yhF+6roHmbBOvqrLiXJcqSZKkBcjQLGneGRge5an9XezckwTlo6eTJl5vXFnDp5o3sT2d4o0ra2ziJUmSpFlnaJY0Lxw5dXbiaPKTL3cxODJGeUkhd2+q47feuommLSlS1TbxkiRJ0twyNEvKidGxyAuHT9LSlqGlLUPb8R4AGmvL+eAtjWzfmuKWdbWUFhXmuFJJkiRdywzNkubMqf6kiVdrtonXyf5hCgsCb1m7hC/en6Y53cCG+gpC8LRrSZIkzQ+GZkmzJsbIvkzvxLXJzx8+yehYpLaihKYtKZq3prh7Uz01i2ziJUmSpPnJ0CzpqhoYHuXpAydobcuwc0+GI6fOArB1eTWfvHcDTekUN65eTKFNvCRJkpQHDM2SXrfjpwey1yZ38OTLJzg7PEpZcQF3bazjN5s20pSuZ3nNolyXKUmSJF02Q7OkyzY6Fvnpa6do2ZM08fr5sTMArFqyiF/dtoqmdIrb1y+lrNgmXpIkScpvhmZJl+TMwDCP7+1iZ1sHu9o76e4borAgcHPjEv7t29Ns35piU6rSJl6SJElaUAzNks4rxsj+zj5as7eEeu6VbkbGIovLi7lvcz1N6RT3bq5ncXlJrkuVJEmSZo2hWdKEwZFRnjnQTUtbhtb2DIdO9AOwpaGKj9+znu3pFDc1LrGJlyRJkq4ZhmbpGpc5M0Bre9Lp+omXu+gfGqW0qIA7NizlN+5eT3M6xcrFNvGSJEnStSlvQnMIoRT4GvBBYBGwE3goxvjaBbb5CvB704Y7YozLZqtOab4bG4u8eOT0RLfrl44kTbxW1JTxyzetpDmd4o4NdSwqsYmXJEmSlDehGfgm8N+ThOYTwNeB/xZCuDnGOHqB7dqB+6Y8v9BcaUHqGRjmiX1d7GzLsKu9k67eQQoC3NS4hM+9bQvN6RTpZVU28ZIkSZKmyYvQHEKoAf4V8LEY4z9nxz4MHALeCnz/ApuPxBiPz36V0vxysKuPnXs6aG3P8OzBboZHI9VlRdy7JUVzup57N6eorbCJlyRJknQheRGagZuBYuAH4wMxxldDCHuAO7hwaF4fQjgKDALPAF+MMR6YzWKlXBgaGeO5V7rZuSdp4nWwqw+ATalKfv3OdTSnU9y8ZglFhQU5rlSSJEnKH/kSmpeRnFbdNW28I/vaTJ4BHgDagBTwZeCpEML1McYT0yeHEB4EHgRobGx8/VVLs6yzZ5DW9gytbRke39dF7+AIJUUF3L5+KQ/csZbmdIrVteW5LlOSJEnKWzkNzSGErwJfusi0pit9/xjj96b9vB8BB4CPklwTPX3+d4DvAGzbti1e6c+VZsvYWORnR89MNPH66WunAWioLuVdN6ygOZ3izo1LKS/Jl9+HSZIkSfNbrr9ZfxP4q4vMOQzcBhQCdUDnlNcagMcv9YfFGHtDCD8DNl1mnVLO9A2O8MTLXbRkT7vO9AwSAty4ejGf2bGZ5q0prltebRMvSZIkaRbkNDTHGLv4xVOuf0EI4XlgGNgB/E12bBWwFXjqUn9eCKEMSAOtV1KvNFcOnejLHk3O8MyBboZGx6gqLeKeLfU0b0lx35Z6llaW5rpMSZIkacHL9ZHmSxJjPB1C+M/AH4UQMkzecupF4Ifj80IIbcCfxhj/NPv8a8A/kBytTgH/I1AB/Pnc/gmkCxseHWP3Kydpbc+wc08H+zuTJl4b6iv46B1raE43sG3tEopt4iVJkiTNqbwIzVkPAyPAI8AiYCfwkWn3aN5Ccgr3uFXAf2HytO4fAbfFGA/NScXSBXT3DbGrPcPOtgyP7e2kZ2CEksICbl1fy4duW0NzOsWapRW5LlOSJEm6puVNaI4xDgKfzi4zzQnTnn9gtuuSLlWMkZ8fO0Nr9rTrF149RYxQX1XK/W9YTlM6xV2b6qgszZv/LSVJkqQFz2/n0iw6OzTKky93sbMtw672DMdODwBww6oaHt6+meZ0iutXVFNQYBMvSZIkaT4yNEtX2avd/bS2J0eTn9p/gqGRMSpLi7h7Ux2/vSNp4pWqKst1mZIkSZIugaFZep1GRsf48eFTE/dO3tvRC8C6ugo+nL02+S1raykpsomXJEmSlG8MzdIVONk3xKN7O2lpy/Do3k5Onx2mqCBw6/pa3rdtNc3pFOvrK3NdpiRJkqTXydAsXYIYI+0dPcnR5D0Zfnz4JGMR6ipL2HFdA83pFHdvqqOqrDjXpUqSJEm6igzN0gwGhkd5ev8JdrZ10NrWyZFTZwF4w8pqPtW8ieZ0ijetrLGJlyRJkrSAGZqlKY6eOpu9NjnDU/u7GBgeo7ykkLs21vHp5o00pVM0VNvES5IkSbpWGJp1TRsdi7xw+OREUG473gNAY205H3hLI83pFLesq6WsuDDHlUqSJEnKBUOzrjmn+4d5dF8nLXs6eHRvJyf7kyZe29Yu4Yv3p2lON7ChvoIQPO1akiRJutYZmrXgxRjZl+mdOJr8/KGTjI5FaitKaNqSonlrirs31VOzyCZekiRJks5laNaCNDA8yo8OnJgIyq+dTJp4Xbe8mk/eu4HmrSluWLWYQpt4SZIkSboAQ7MWjOOnB7IhuYMnXz7B2eFRyooLuGtjHQ/dt5GmdD3LaxblukxJkiRJecTQrLw1Ohb56WunaNmTHE3++bEzAKxasohf3baKpnSK29cvtYmXJEmSpCtmaFZeOTMwzGN7O2lpy7CrvZPuviEKCwI3Ny7h8+9I05xOsSlVaRMvSZIkSVeFoVnzWoyR/Z19tLZl2NnWwe5XTjIyFllcXsx9m+tpSqe4d3M9i8tLcl2qJEmSpAXI0Kx5Z3BklGcPdrNzT4bW9gyHTvQDkF5WxcfvWc/2dIqbGpfYxEuSJEnSrDM0a17InBmgtT25NvmJfV30DY1SWlTAHRuW8ht3r6c5nWLlYpt4SZIkSZpbhmblxNhY5F+OnJ64JdS/HDkNwIqaMt5900q2b01x+/o6FpXYxEuSJElS7hiaNWd6BoZ5Yl8XLW0ZWts76eodpCDATY1L+NzbttCcTpFeVmUTL0mSJEnzhqFZs+pgV9/EvZOfPdjN8GikuqyIe7ekaE7Xc+/mFLUVNvGSJEmSND8ZmnVVDY2M8dwr3ROnXR/s6gNgU6qSX79zHc3pFDevWUJRYUGOK5UkSZKkizM063Xr7BmktT1Da1uGx/d10Ts4QklRAbevX8oDd6ylOZ1idW15rsuUJEmSpMtmaNZlizHys6Nn2LknQ0t7hp++egqAhupS3nXDcprTDdy5cSnlJe5ekiRJkvKbqUaXpG9whCde7qI1e9p1pmeQEOCGVYv5zI7NNKVTXL+i2iZekiRJkhYUQ7NmdOhE38S1yc8c6GZodIyq0iLu2VxPczrFvVvqqasszXWZkiRJkjRrDM2aMDw6xu5XTtLS1kFLW4b9nUkTr/X1FXz0jjU0pVO8ZW0txTbxkiRJknSNMDRf4070DrKrvZOW9gyP7e2kZ2CE4sLAbeuX8j/cuobmdIq1dRW5LlOSJEmScsLQfI2JMfLzY2dobcuwsy3DT149RYxQX1XKO96wjOZ0A3dtqqOy1F1DkiRJkkxG14CzQ6M8+XIXO9sy7GrPcOz0AAA3rKrht7ZvYnu6getXVFNQYBMvSZIkSZoqb0JzCOFB4IPATUANsC7G+MolbPde4H8GNgD7gS/FGP+vWSx1XnjtZP/E0eSn9p9gaGSMipJC7tlcz2/vSHHflnpSVWW5LlOSJEmS5rW8Cc1AOfAD4P8BvnEpG4QQbgceAX4P+HvgPcB3Qwh3xhifma1Cc2FkdIwfHz6V7Xbdwd6OXgDWLi3nQ7euYfvWpIlXSZFNvCRJkiTpUuVNaI4xfhMghLDtMjZ7GGiNMf777PN/H0Joyo5/8CqXOOdO9Q/x6N5Odu7J8OjeTk6fHaaoIHDLulret201TekUG+orc12mJEmSJOWtvAnNV+h24E+mjX0f+FQOarlqjpw6y8P/9QWeP3SSsQhLK0rYcV0DzekUd22qo7qsONclSpIkSdKCsNBD8zKgY9pYR3b8F2Svm34QoLGxcXYrex3qKksA+FTTRpq3NvCmlTU28ZIkSZKkWZDT0BxC+CrwpYtMa4ox7pqDcogxfgf4DsC2bdviXPzMK1FaVMh3P3FHrsuQJEmSpAUv10eavwn81UXmHH4d738caJg21pAdlyRJkiTpgnIammOMXUDXLP6Ip4EdwB9PGdsBPDWLP1OSJEmStEDk+kjzJQshLCO5Fnlzdui6EMJi4HCMsTs7ZyfwbIzxC9k53wIeCyF8Hvi/gV8GmoC75rR4SZIkSVJeyqeb9n4CeAH46+zzf8w+/++mzNkALB9/EmN8CvgA8ADwIvAR4P0L7R7NkiRJkqTZEWKct/2ucmrbtm1x9+7duS5DkiRJkjQLQgjPxxi3XWxePh1pliRJkiRpThmaJUmSJEmagaFZkiRJkqQZGJolSZIkSZqBoVmSJEmSpBkYmiVJkiRJmoGhWZIkSZKkGRiaJUmSJEmaQYgx5rqGeSmE0AkcynUdF1EHdOW6CM1L7huaifuGZuK+oQtx/9BM3Dc0k3zYN9bEGOsvNsnQnMdCCLtjjNtyXYfmH/cNzcR9QzNx39CFuH9oJu4bmslC2jc8PVuSJEmSpBkYmiVJkiRJmoGhOb99J9cFaN5y39BM3Dc0E/cNXYj7h2bivqGZLJh9w2uaJUmSJEmagUeaJUmSJEmagaFZkiRJkqQZGJrzRAjhwRBCawjhVAghhhDWXuJ27w0h/DyEMJhd//LsVqq5FkIoDSH8SQihK4TQF0L4f0MIqy6yzVey+9HU5fhc1azZE0J4KIRwMIQwEEJ4PoRw90Xm35udNxBCOBBC+MRc1aq5dTn7RgjhvvN8RsQQQnoua9bsCyHck/1340j27/iBS9jmjSGER0MIZ7Pb/U8hhDAH5WoOXe6+EUJYO8PnxtvnqGTNkRDCF0IIz4UQzoQQOkMI/xBCeMMlbJe3nx2G5vxRDvwA+MqlbhBCuB14BPhr4Mbs+rshhFtno0DlzDeB9wIfBO4GqoH/FkIovMh27cDyKcsbZ7NIzb4QwvuBbwH/C3AT8BTwvRBC4wzz1wH/lJ13E/AHwJ+EEN47NxVrrlzuvjHF9Zz7ObFvNutUTlQCLwG/BZy92OQQQjXwz0AH8Jbsdp8DfmcWa1RuXNa+McXbOfdzo+Xql6Ycuw/4T8AdQDMwAvwwhFA70wb5/tlhI7A8E0LYBjwHrIsxvnKRuY8AtTHGHVPGfgh0xhg/OKuFak6EEGqATuBjMca/zo6tBg4B74gxfn+G7b4C/EqM8aK/FVT+CCE8A7wYY/z4lLF9wN/GGL9wWsyZhQAABlFJREFUnvl/CLwnxrhpytj/AVwfY7x9LmrW3LiCfeM+oBWojzF2zVmhyqkQQi/wqRjjn11gzieBPwQaYoxns2NfBj4JrIp+sVyQLnHfWAscBN4SY9w9N5VpPgghVAKngXfHGP9hhjl5/dnhkeaF7XaSo9NTfZ/kt0JaGG4Gipny9xxjfBXYw8X/nteHEI5mT9f8ryGE9bNYp2ZZCKGEZH+Y/v/8D5h5X5jpM2JbCKH46laoXLnCfWPc7hDCsRDCzhBC06wUqHxzO/D4+JferO8DK4C1OalI883fhxAyIYQnQwi/kutiNCeqSHLlyQvMyevPDkPzwraM5BSIqTqy41oYlgGjwPQjQRf7e34GeIDkFKqPZ+c+FUJYOgs1am7UAYVc3v/zM31GFGXfTwvDlewbx0h++/9e4D0kl3PsvNg18romzPS5Mf6arl29wGeB9wH3AzuBR0IIH8ppVZoL3wJ+Ajx9gTl5/dlRlOsCrmUhhK8CX7rItKYY4645KEfzyKXuG1f6/jHG7037eT8CDgAfBb5+pe8raWGIMbaTBOVxT2dPvfwc8HguapI0v2Uv5fgPU4Z2hxDqgN8F/io3VWm2hRC+DtwF3BVjHM11PbPF0Jxb3+TiHyKHX8f7Hwcapo01ZMc1v13qvnEbyRGkOpJrm8c1cBlfbGOMvSGEnwGbLjpZ81UXyVkHl/P//EyfESP84tkLyl9Xsm+czzPAB65WUcpbM31ujL8mTfUM8LFcF6HZEUL4Bsm/C00xxgMXmZ7Xnx2G5hzK/kZuNr+YPg3sAP54ytgOkq6pmscudd8IITwPDJP8vf5NdmwVsJXL+HsOIZQBaZLGP8pDMcah7P6wA/julJd2AH83w2ZPA9NvQ7cD2B1jHL76VSoXrnDfOJ8bSU7b1rXtaeAPQwhlMcaB7NgO4CjwSs6q0nzl58YCFUL4FvB+ksDcdgmb5PVnh9c054kQwrIQwo3A5uzQdSGEG6e2ds82avmDKZt9C2gOIXw+hJAOIXyB5JTeb85d5ZpNMcbTwH8G/iiE8NYQwk3AXwIvAj8cnxdCaAshfGrK86+F5P6867K3IPtboAL487n9E+gq+zrwQAjhN0IIW7P/oK0Avg0QQviLEMJfTJn/bWBlCOGb2fm/QXKt+9fmunDNusvaN0IID4cQ3h1C2BRCuD77b8u7gT/NSfWaNSGEyuz3iRtJvhc2Zp83Zl//gxDCzimb/A3QD/xZCOENIYT3AJ8Hvj7fu9/q8lzuvhFC+GgI4deynzFbQgifBX4T+JPc/Ak0W0II/5HkDIJfA05mc8qybBft8TkL6rPDI8354xPA7015/o/Z9ceAP8s+3gC8Oj4hxvhUCOEDwFeB3wf2A++PMT4z69VqLj1McjrtI8AiksYbH5l2XckWzm3stAr4L0ye1v0j4LYY46E5qVizIsb4SLaZ25dJ7o35EnD/lL/XxmnzD4YQ7ge+QdL06Sjwb2KMl3P0UXngcvcNoITkLKVVJPdn/RnwzhjjP81RyZo72zj3LKN/l13+nOSXaMtJvl8AyS9rQwg7gP8I7CbplvsfsB/GQnRZ+0bWl4E1JJeE7AV+Pcbo9cwLz0PZ9c5p4/8O+Er28YL67PA+zZIkSZIkzcDTsyVJkiRJmoGhWZIkSZKkGRiaJUmSJEmagaFZkiRJkqQZGJolSZIkSZqBoVmSJEmSpBkYmiVJkiRJmoGhWZIkSZKkGRiaJUmSJEmagaFZkqRrVAjhV0IIgyGENVPGvhVC2B9CaMhlbZIkzRchxpjrGiRJUg6EEALwHPBCjPHjIYTPAr8L3Blj3Jfb6iRJmh+Kcl2AJEnKjRhjDCF8EfjHEMJ+4IvAdgOzJEmTPNIsSdI1LoTwFHAL8K4Y4/dyXY8kSfOJ1zRLknQNCyE0AzcAAejIcTmSJM07HmmWJOkaFUK4AXgM+G3gnUBljPFtua1KkqT5xdAsSdI1KNsx+2ng2zHG3w8hvAF4EWiOMe7KaXGSJM0jhmZJkq4xIYRa4EngsRjjv54y/gjQGGO8PWfFSZI0zxiaJUmSJEmagY3AJEmSJEmagaFZkiRJkqQZGJolSZIkSZqBoVmSJEmSpBkYmiVJkiRJmoGhWZIkSZKkGRiaJUmSJEmagaFZkiRJkqQZGJolSZIkSZrB/w9G8LNNF8Qx8QAAAABJRU5ErkJggg==\n",
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