MAT120_LAB_Lec06.ipynb

{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "provenance": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Forming system of differential Equation from physical problem"
      ],
      "metadata": {
        "id": "9oG_eJw3Zprz"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Problem\n",
        "A ball is released from a height in three different fluids with drag coefficient $b=1kgs^-1,1.5kgs^-1$ and  $2kgs^-1$. Drag force is directly proportional to the instantaneous velocity of the ball. Determine the position (vertical distance) of the ball at each second from $t=0s$ to $t=100s$. Take,\n",
        " $gravity, g=9.8ms^-2$ ;  $mass, m=1kg$"
      ],
      "metadata": {
        "id": "mSh6Q56yWeuM"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Modelling\n",
        "![Add a 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)\n"
      ],
      "metadata": {
        "id": "gOKoPvSrcdRy"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "Let, $y(t)$ is vertical position of the ball at time $t$.\n",
        "\n",
        "$v(t)$ is the vertical velocity of the ball at time $t$.\n",
        "\n",
        "Here, Drag force is directly proportional to instantaneous velocity of the ball.\n",
        "\n",
        "$F_D∝v$\n",
        "\n",
        "$F_D=bv$\n",
        "\n",
        "The gravity force exerts on the ball:\n",
        "\n",
        "$F_G=mg$\n",
        "\n",
        "Net force exerts on the ball:\n",
        "\n",
        "$∑F=F_G-F_D$\n",
        "\n",
        "According to Newton's second law of motion,\n",
        "$∑F=ma$.\n",
        "\n",
        "So, $ma=F_G-F_D$\n",
        "\n",
        "or, $ma=mg-bv$\n",
        "\n",
        "or, $a=g-\\frac{b}{m}v$\n",
        "\n",
        "or, $d^2y/dt^2=g-\\frac{b}{m}dy/dt$\n",
        "\n",
        "This is a second order differential equation.\n",
        "\n"
      ],
      "metadata": {
        "id": "VmnWGQw6fPFX"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Convert the 2nd order ODE into two first order System of differential equations"
      ],
      "metadata": {
        "id": "epYFj6F_gV_q"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "$dy/dt=v$\n",
        "\n",
        "$d^2y/dt^2=dv/dt$\n",
        "\n",
        "The second order ode become,\n",
        "\n",
        "$dv/dt=g-\\frac{b}{m}v$\n",
        "\n",
        "Two first order ODEs of the physical system are now,\n",
        "\n",
        "{$dy/dt=v$ ,\n",
        "\n",
        "$dv/dt=g-\\frac{b}{m}v$}\n"
      ],
      "metadata": {
        "id": "EsjmPx4qlOoV"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Solving Using odeint\n",
        "Scipy **odeint** function is designed to handle first oder system of differential equations. Let's see how can we solve the system using **odeint**."
      ],
      "metadata": {
        "id": "76zPBk3rnIuU"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def falling_body(y,t,drag_coff):\n",
        "  # unpack tuple y\n",
        "  y,v=y\n",
        "  gravity=9.8\n",
        "  mass=1\n",
        "  dydt=v\n",
        "  dvdt=gravity-(drag_coff/mass)*v\n",
        "  return [dydt, dvdt]"
      ],
      "metadata": {
        "id": "c_rKtInJnD43"
      },
      "execution_count": 54,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "Initial conditions of the system:\n",
        "$y(0)=0$\n",
        "\n",
        "$y'(0)=v=0$"
      ],
      "metadata": {
        "id": "R3MZABhopFSJ"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from scipy.integrate import odeint\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "# Initial conditions\n",
        "ini_conds=[0,0]\n",
        "# Time at which positions will be determined\n",
        "step= np.linspace(0,101,100)\n",
        "drag_coff=[1,1.5,2]\n",
        "result=[]\n",
        "for drag in drag_coff:\n",
        "  # Drag coefficient is passed as an additional argument in a tuple\n",
        "  soln=odeint(falling_body,ini_conds,step,args=(drag,))\n",
        "  '''Odeint returns a list of lists where each list contains the solution of both odes. The first one is the position at time t\n",
        "     and second one is the velocity at time t'''\n",
        "  print(f'The position of the object at different time for drag coefficient {drag} is:{soln[:,0]}')\n",
        "  result.append(soln[:,0])\n"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6FtlSp2ynDkV",
        "outputId": "60f53a06-468c-4eb9-d280-a2a9d744429c"
      },
      "execution_count": 55,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "The position of the object at different time for drag coefficient 1 is:[  0.           3.73109637  11.46972624  20.65316046  30.35747855\n",
            "  40.24958681  50.20939757  60.19361657  70.1866353   80.18282653\n",
            "  90.18016152 100.17790884 110.17580481 120.17375441 130.17172334\n",
            " 140.16969918 150.16767755 160.16565687 170.16363648 180.1616162\n",
            " 190.15959598 200.15757576 210.15555555 220.15353535 230.15151515\n",
            " 240.14949495 250.14747474 260.14545454 270.14343434 280.14141414\n",
            " 290.13939394 300.13737374 310.13535353 320.13333333 330.13131313\n",
            " 340.12929293 350.12727273 360.12525252 370.12323232 380.12121212\n",
            " 390.11919192 400.11717172 410.11515151 420.11313131 430.11111111\n",
            " 440.10909091 450.10707071 460.1050505  470.1030303  480.1010101\n",
            " 490.0989899  500.0969697  510.09494949 520.09292929 530.09090909\n",
            " 540.08888889 550.08686869 560.08484848 570.08282828 580.08080808\n",
            " 590.07878788 600.07676768 610.07474747 620.07272727 630.07070707\n",
            " 640.06868687 650.06666667 660.06464646 670.06262626 680.06060606\n",
            " 690.05858586 700.05656566 710.05454545 720.05252525 730.05050505\n",
            " 740.04848485 750.04646465 760.04444444 770.04242424 780.04040404\n",
            " 790.03838384 800.03636364 810.03434343 820.03232323 830.03030303\n",
            " 840.02828283 850.02626263 860.02424242 870.02222222 880.02020202\n",
            " 890.01818182 900.01616162 910.01414141 920.01212121 930.01010101\n",
            " 940.00808081 950.00606061 960.0040404  970.0020202  980.        ]\n",
            "The position of the object at different time for drag coefficient 1.5 is:[  0.           3.25261166   9.17918242  15.68458521  22.31528781\n",
            "  28.97311406  35.63681179  42.30178052  48.96702436  55.63232778\n",
            "  62.29764408  68.96296318  75.62828288  82.29360271  88.95892256\n",
            "  95.62424242 102.28956229 108.95488215 115.62020202 122.28552188\n",
            " 128.95084175 135.61616162 142.28148148 148.94680135 155.61212121\n",
            " 162.27744108 168.94276094 175.60808081 182.27340067 188.93872054\n",
            " 195.6040404  202.26936027 208.93468013 215.6        222.26531987\n",
            " 228.93063973 235.5959596  242.26127946 248.92659933 255.59191919\n",
            " 262.25723906 268.92255892 275.58787879 282.25319865 288.91851852\n",
            " 295.58383838 302.24915825 308.91447811 315.57979798 322.24511785\n",
            " 328.91043771 335.57575758 342.24107744 348.90639731 355.57171717\n",
            " 362.23703704 368.9023569  375.56767677 382.23299663 388.8983165\n",
            " 395.56363636 402.22895623 408.89427609 415.55959596 422.22491582\n",
            " 428.89023569 435.55555556 442.22087542 448.88619529 455.55151515\n",
            " 462.21683502 468.88215488 475.54747475 482.21279461 488.87811448\n",
            " 495.54343434 502.20875421 508.87407407 515.53939394 522.2047138\n",
            " 528.87003367 535.53535354 542.2006734  548.86599327 555.53131313\n",
            " 562.196633   568.86195286 575.52727273 582.19259259 588.85791246\n",
            " 595.52323232 602.18855219 608.85387205 615.51919192 622.18451178\n",
            " 628.84983165 635.51515152 642.18047138 648.84579125 655.51111111]\n",
            "The position of the object at different time for drag coefficient 2 is:[  0.           2.86743154   7.58936964  12.55234939  17.54665883\n",
            "  22.54504038  27.54395121  32.54293083  37.54191939  42.54090912\n",
            "  47.53989899  52.53888889  57.53787879  62.53686869  67.53585859\n",
            "  72.53484848  77.53383838  82.53282828  87.53181818  92.53080808\n",
            "  97.52979798 102.52878788 107.52777778 112.52676768 117.52575758\n",
            " 122.52474747 127.52373737 132.52272727 137.52171717 142.52070707\n",
            " 147.51969697 152.51868687 157.51767677 162.51666667 167.51565657\n",
            " 172.51464646 177.51363636 182.51262626 187.51161616 192.51060606\n",
            " 197.50959596 202.50858586 207.50757576 212.50656566 217.50555556\n",
            " 222.50454545 227.50353535 232.50252525 237.50151515 242.50050505\n",
            " 247.49949495 252.49848485 257.49747475 262.49646465 267.49545455\n",
            " 272.49444444 277.49343434 282.49242424 287.49141414 292.49040404\n",
            " 297.48939394 302.48838384 307.48737374 312.48636364 317.48535354\n",
            " 322.48434343 327.48333333 332.48232323 337.48131313 342.48030303\n",
            " 347.47929293 352.47828283 357.47727273 362.47626263 367.47525253\n",
            " 372.47424242 377.47323232 382.47222222 387.47121212 392.47020202\n",
            " 397.46919192 402.46818182 407.46717172 412.46616162 417.46515152\n",
            " 422.46414141 427.46313131 432.46212121 437.46111111 442.46010101\n",
            " 447.45909091 452.45808081 457.45707071 462.45606061 467.45505051\n",
            " 472.4540404  477.4530303  482.4520202  487.4510101  492.45      ]\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Plot position against time for each drag coefficient\n",
        "for i,drag in enumerate(drag_coff):\n",
        "  plt.plot(step,result[i],label=f\"drag={drag}\")\n",
        "  plt.xlabel(xlabel='time')\n",
        "  plt.ylabel(ylabel='position')\n",
        "  plt.legend()"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 449
        },
        "id": "ZYjewdfLppTt",
        "outputId": "34aef81d-b58d-492c-9376-24f14a339bf6"
      },
      "execution_count": 56,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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\n"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Solving system of differential equations using sympy"
      ],
      "metadata": {
        "id": "RIBsXSpcnH2W"
      }
    },
    {
      "cell_type": "code",
      "execution_count": 57,
      "metadata": {
        "id": "Me8pO9NOWWU_",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 87
        },
        "outputId": "88d815bd-214b-4268-bfcf-fd22a135952d"
      },
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Eq(Derivative(y(t), t), v(t))"
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d t} y{\\left(t \\right)} = v{\\left(t \\right)}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Eq(Derivative(v(t), t), 9.8 - v(t))"
            ],
            "text/latex": "$\\displaystyle \\frac{d}{d t} v{\\left(t \\right)} = 9.8 - v{\\left(t \\right)}$"
          },
          "metadata": {}
        }
      ],
      "source": [
        "import sympy as sp\n",
        "from sympy import *\n",
        "t=symbols('t')\n",
        "y=Function('y')\n",
        "v=Function('v')\n",
        "Eq1=Eq((diff(y(t),t,1)),v(t))\n",
        "# Let's say drag_coefficient 1\n",
        "Eq2=Eq(diff(v(t),t,1),9.8-Rational(1,1)*v(t))\n",
        "display(Eq1)\n",
        "display(Eq2)\n"
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Define the system as a list of differential equations\n",
        "system=[Eq1,Eq2]"
      ],
      "metadata": {
        "id": "rEvs2fwzwvM_"
      },
      "execution_count": 58,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# Define the intial conditions\n",
        "boundary={y(t).subs(t,0):'0',sp.diff(y(t),t,1).subs(t,0):'0'}"
      ],
      "metadata": {
        "id": "ZgYuv3L3xCRy"
      },
      "execution_count": 59,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "sp.dsolve(system)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "8jL6TUsyxc_J",
        "outputId": "45b041cc-d4b9-496b-f2d9-e3a830aa61b3"
      },
      "execution_count": 60,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[Eq(y(t), 1.0*C1 - C2*exp(-t) + 9.8*t - 9.8), Eq(v(t), C2*exp(-t) + 9.8)]"
            ]
          },
          "metadata": {},
          "execution_count": 60
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "soln=sp.dsolve(system,ics=boundary)"
      ],
      "metadata": {
        "id": "D25iocGlxm6Q"
      },
      "execution_count": 61,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "for i in range(2):\n",
        "  display(soln[i])"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 61
        },
        "id": "nRq8QHAnxqmO",
        "outputId": "9522a28a-195c-424b-9400-f14035d10609"
      },
      "execution_count": 62,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Eq(y(t), 9.8*t - 9.8 + 9.8*exp(-t))"
            ],
            "text/latex": "$\\displaystyle y{\\left(t \\right)} = 9.8 t - 9.8 + 9.8 e^{- t}$"
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "Eq(v(t), 9.8 - 9.8*exp(-t))"
            ],
            "text/latex": "$\\displaystyle v{\\left(t \\right)} = 9.8 - 9.8 e^{- t}$"
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# Apparently we got same solution for drag coefficient 1 as we got while using odeint.\n",
        "sp.plot(soln[0].rhs,(t,0,100))"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 501
        },
        "id": "8ccxdRQQxz1Y",
        "outputId": "14dbc5f2-324a-4c2d-f5e5-5b14a75d9b94"
      },
      "execution_count": 53,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/plain": [
              "<Figure size 640x480 with 1 Axes>"
            ],
            "image/png": 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TSU9PByA9PZ3Q0NCaEgeQlJSE1Wpl+/bt3/m8FRUV2Gy2Wg8RERFntP9sEXfM38La/bl4eVj404ieLE5OUIlr4lxioXzatGnYbDa6d++Oh4cHdrudF154geTkZAByc7/5L4+IiIhaXxcREVGzLzc3l/Dw8Fr7PT09CQsLqxnzbbNnz+bZZ5+t78MRERGpN4ZhsHR7Ds+tPkil3UHrUD8WJScQ1zbU7GjSCFyiyL377rssXbqUtLQ0evXqRVZWFpMmTSIqKorx48c32PNOnz6dKVOm1PzZZrPRtm3bBns+ERGRq1FSUc30D/axes85AJJ6hPPy6FhC/b1NTiaNxSWK3BNPPMG0adMYM2YMAL179+bUqVPMnj2b8ePHExkZCUBeXh6tWrWq+bq8vDzi4uIAiIyMJD8/v9b3ra6upqCgoObrv83HxwcfH58GOCIREZFrc+i8jZSlGRy/WIqH1cK0od357Y3RuqDBzbjEe+TKysqwWmtH9fDwwOFwABAdHU1kZCQbNmyo2W+z2di+fTsDBgwAYMCAARQWFrJ79+6aMRs3bsThcJCYmNgIRyEiInLtDMNgxc4c7l60leMXS2kV4su7v7ueh27qqBLnhlzijNyIESN44YUXaNeuHb169SIzM5NXX32VBx54AACLxcKkSZOYNWsWXbp0ITo6mhkzZhAVFcXdd98NQI8ePRg6dCgPPfQQS5YsoaqqitTUVMaMGVOnK1ZFRETMVlZZzdOr9vNB5lkAbunWklfviyMsQEup7solbj9SXFzMjBkzWLVqFfn5+URFRTF27FieeeYZvL2/+cdrGAYzZ87kjTfeoLCwkEGDBrF48WK6du1a830KCgpITU1l9erVWK1WRo0axfz58wkMDKxTDt1+REREzHI0r5gJSzM4ml+C1QJTB3fjkZs7YbXqLJw7c4ki5yxU5ERExAwfZJzhqVX7uVJlJzzIh/lj47m+Y3OzY4kTcImlVREREXdUXmVn5kcHWLHrNACDOrfgtfvjaBmkC/HkGypyIiIiTuj4hRImLM3gcG4xFgs8dlsXJv6sCx5aSpX/oCInIiLiZD7ec47p7++ltNJOi0Bv5t0fz6AuLcyOJU5IRU5ERMRJlFfZmfXJQd7ZlgNA/+gwFoyNJyLY1+Rk4qxU5ERERJzAqUulTFiawYFz33yud8qtnZic1BVPD5e45auYREVORETEZOv2n+eJlXsprqimmb8Xr90fxy3dwn/8C8XtqciJiIiYpLLawey1h3hr60kA+rZvxoKx8USF+pkbTFyGipyIiIgJzlwuIyUtkz2nCwH43U0deXxIN7y0lCpXQUVORESkkX12MI+pK/dQdKWKED8vXhkdS1LPCLNjiQtSkRMREWkkVXYHL3+azV83Hwcgtm0oC8fG0zbM3+Rk4qpU5ERERBrB+aIrpKZlsvvUZQB+M7AD02/vgbenllLlp1ORExERaWBfZOczeUUWl8uqCPLxZO69fbi9dyuzY0kToCInIiLSQKrtDuZ9dpRFXxzDMCCmdTCLxiXQvnmA2dGkiVCRExERaQD5tnImLstk+4kCAH5xfTueHt4TXy8Pk5NJU6IiJyIiUs+2HrvIY8szuVhSSYC3B7NH9eHO2CizY0kTpCInIiJST+wOgwUbj/KXDUcxDOgeGcSi5AQ6tQw0O5o0USpyIiIi9eBiSQWTlmex5dhFAMZc15Y/3dlLS6nSoFTkRERErtH245eYuCyT/OIK/Lw8eOGeGEYmtDE7lrgBFTkREZGfyOEweH3T17zyr2wcBnQJD2RxcgJdIoLMjiZuQkVORETkJygorWTKu1l8kX0BgJHxrZl1Twz+3npplcajf20iIiJXafepAlLTMjlfVI6Pp5Xn7urFff3aYrFYzI4mbkZFTkREpI4Mw+BvX57gpXWHqXYYdGwRwKLkBHq0CjY7mrgpFTkREZE6KCqrYurKPXx2KA+AEbFRzB7Zm0AfvZSKefSvT0RE5EfsOV1ISloGZy5fwdvDyjMjepKc2E5LqWI6FTkREZHvYRgGb391khf/eYgqu0G7MH8WJycQ0zrE7GgigIqciIjId7KVV/Hke3tZuz8XgKG9Ipk7ug/Bvl4mJxP5/6nIiYiIfMv+s0WkpGVw6lIZXh4W/jisB7++oYOWUsXpqMiJiIj8H8MwWLo9h+fWHKSy2kHrUD8WJScQ1zbU7Ggi30lFTkREBCipqOaPH+zj4z3nALiteziv3BdLqL+3yclEvp+KnIiIuL3DuTYmvJPB8YuleFgtPDm0Gw/d2FFLqeL0VORERMStvbvrNM98tJ/yKgeRwb4sHBdPvw5hZscSqRMVORERcUtlldXM+PAA72ecAeCmri157b5Ymgf6mJxMpO5U5ERExO0cyy/mkXcyOJpfgtUCUwd345GbO2G1ailVXIuKnIiIuJVVmWd4atV+yirttAzyYf6YeAZ0am52LJGfREVORETcQnmVnWdXH2DZjtMADOzcnHn3x9MySEup4rpU5EREpMk7fqGElLRMDp23YbHAoz/rwqO3dcFDS6ni4lTkRESkSVuz9xzT3t9HSUU1zQO8mTcmjhu7tDQ7lki9UJETEZEmqaLazqw1h/jfbacA6B8dxoKx8UQE+5qcTKT+qMiJiEiTk3OpjJS0DPadLQJgwi2dmPLzrnh6WE1OJlK/VORERKRJWbc/lyfe20NxeTWh/l68dn8ct3YLNzuWSINQkRMRkSahstrBnLWH+X9bTwCQ0C6UheMSiAr1MzmZSMNRkRMREZd35nIZqWmZZJ0uBOChG6P5w9DueGkpVZo4FTkREXFpGw7lMeXdPRRdqSLY15OXR8cyuFek2bFEGoWKnIiIuKQqu4OXP83mr5uPAxDbJoSF4xJoG+ZvcjKRxqMiJyIiLud80RUmpmWy69RlAH59QwemD+uOj6eHyclEGpeKnIiIuJRNRy4weUUWBaWVBPp4MvfePgzr3crsWCKmUJETERGXYHcYzPvsCAs/P4ZhQM9WwSxOTqBDiwCzo4mYRkVOREScXr6tnEeXZ7LteAEAyYntmHFHT3y9tJQq7k1FTkREnNpXxy7y6PIsLpZU4O/tweyRvbkrrrXZsUScgoqciIg4JbvDYOHGY8zbcATDgG4RQSxKTqBzeKDZ0USchoqciIg4nYslFUxekcWXRy8CcF+/Njx7Zwx+3lpKFflPKnIiIuJUth+/xKPLM8mzVeDrZWXW3b25t28bs2OJOCUVORERcQoOh8GSzV/zyr+OYHcYdA4PZHFyAl0jgsyOJuK0VORERMR0l0srmfJuFp9nXwBgZHxrnr87hgAfvUyJ/BD9hIiIiKl2n7rMxLQMzhWV4+Np5dk7e3H/dW2xWCxmRxNxeipyIiJiCsMw+PuWE8xZe5hqh0F0iwAWjUugZ1Sw2dFEXIaKnIiINLqisioef28P6w/mATC8TyvmjOxNkK+XyclEXIuKnIiINKo9pwtJScvgzOUreHtYmXFHD35xfXstpYr8BCpyIiLSKAzD4H/STzHrk4NU2Q3ahvmxaFwCfdqEmh1NxGWpyImISIOzlVcx/f19fLLvPABDekUw995YQvy0lCpyLVTkRESkQR04V0TK0gxOXirD02ph+rAePDCwg5ZSReqBipyIiDQIwzBYtuM0f1p9gMpqB61D/Vg4Lp74ds3MjibSZKjIiYhIvSutqOaPq/bxUdY5AG7rHs4r98US6u9tcjKRpkVFTkRE6lV2bjETlu7m6wuleFgt/GFINx66sSNWq5ZSReqbipyIiNSblbtOM+Oj/ZRXOYgM9mXhuHj6dQgzO5ZIk6UiJyIi1+xKpZ0ZH+3nvd1nALipa0teuy+W5oE+JicTadpU5ERE5Jocyy8hZWkG2XnFWC0w5eddmXBLZy2lijQCFTkREfnJPsw8yx9X7aOs0k7LIB/mj4lnQKfmZscScRsqciIictXKq+w8u/ogy3bkAHBDp+b8ZUw8LYO0lCrSmKxmB6irs2fP8otf/ILmzZvj5+dH79692bVrV81+wzB45plnaNWqFX5+fiQlJXH06NFa36OgoIDk5GSCg4MJDQ3lwQcfpKSkpLEPRUTEpZ24WMo9i79i2Y4cLBZ47LYu/O+DiSpxIiZwiSJ3+fJlBg4ciJeXF2vXruXgwYO88sorNGv2/99Ucu7cucyfP58lS5awfft2AgICGDJkCOXl5TVjkpOTOXDgAOvXr2fNmjVs3ryZhx9+2IxDEhFxSZ/sPc+IBVs4dN5G8wBv/ueB/kz+eVc89H44EVNYDMMwzA7xY6ZNm8bWrVv58ssvv3O/YRhERUUxdepUHn/8cQCKioqIiIjg7bffZsyYMRw6dIiePXuyc+dO+vXrB8C6desYNmwYZ86cISoq6kdz2Gw2QkJCKCoqIjg4uP4OUETEyVVU23nhk0P8T/opAPp3CGPBuHgign1NTibi3lzijNzHH39Mv379GD16NOHh4cTHx/Pmm2/W7D9x4gS5ubkkJSXVbAsJCSExMZH09HQA0tPTCQ0NrSlxAElJSVitVrZv3/6dz1tRUYHNZqv1EBFxNzmXyrj39fSaEvfILZ1IeyhRJU7ECbhEkTt+/Divv/46Xbp04dNPP+WRRx7h0Ucf5R//+AcAubm5AERERNT6uoiIiJp9ubm5hIeH19rv6elJWFhYzZhvmz17NiEhITWPtm3b1vehiYg4tU8P5DJ8wZfsO1tEqL8Xb/36Op4c2h1PD5d4+RBp8lziqlWHw0G/fv148cUXAYiPj2f//v0sWbKE8ePHN9jzTp8+nSlTptT82WazqcyJiFuorHbw0rrD/H3LCQDi24WycFwCrUP9TE4mIv/JJf6TqlWrVvTs2bPWth49epCT881l75GRkQDk5eXVGpOXl1ezLzIykvz8/Fr7q6urKSgoqBnzbT4+PgQHB9d6iIg0dWcLr3DfX9NrStxvB0Wz4uEBKnEiTsglitzAgQPJzs6ute3IkSO0b98egOjoaCIjI9mwYUPNfpvNxvbt2xkwYAAAAwYMoLCwkN27d9eM2bhxIw6Hg8TExEY4ChER57fxcB7D539J1ulCgnw9+esv+/L0HT3x9nSJlwsRt+MSS6uTJ0/mhhtu4MUXX+S+++5jx44dvPHGG7zxxhsAWCwWJk2axKxZs+jSpQvR0dHMmDGDqKgo7r77buCbM3hDhw7loYceYsmSJVRVVZGamsqYMWPqdMWqiEhTVm138PK/jrBk09cA9GkTwqJxCbQN8zc5mYj8EJe4/QjAmjVrmD59OkePHiU6OpopU6bw0EMP1ew3DIOZM2fyxhtvUFhYyKBBg1i8eDFdu3atGVNQUEBqaiqrV6/GarUyatQo5s+fT2BgYJ0y6PYjItIU5RaVM3FZBjtPXgbg1zd0YPqw7vh4epicTER+jMsUOWegIiciTc3mIxeYtCKLgtJKAn08eWlUH4b3aWV2LBGpI5dYWhURkfpldxj85bMjLPj8GIYBPVsFszg5gQ4tAsyOJiJXQUVORMTN5BeX89iyLNKPXwJgXGI7nrmjJ75eWkoVcTUqciIibuSrry/y6LIsLpZU4O/tweyRvbkrrrXZsUTkJ1KRExFxAw6HwcLPjzHvsyM4DOgWEcSi5AQ6h9ftYi8RcU4qciIiTdylkgomrcjiy6MXARjdtw3P3RWDn7eWUkVcnYqciEgTtuNEAROXZZBnq8DXy8rzd8Uwup8+alCkqVCRExFpghwOg79uPs7L/8rG7jDo1DKAxcl96RYZZHY0EalHKnIiIk3M5dJKpq7cw8bD33y+9N1xUbxwT28CfPQrX6Sp0U+1iEgTkpFzmdSlGZwrKsfb08qzd/ZizHVtsVgsZkcTkQagIici0gQYhsHft5xgztrDVDsMOjT3Z1FyAr2iQsyOJiINSEVORMTFFV2p4g/v7eHTA3kADO/dijmjehPk62VyMhFpaCpyIiIubO+ZQlLSMjhdcAVvDytP39GDX17fXkupIm5CRU5ExAUZhsH/bjvFrDWHqLQ7aBvmx6JxCfRpE2p2NBFpRCpyIiIupri8imnv7+OTfecBGNwzgj+PjiXET0upIu5GRU5ExIUcPGdjwtLdnLxUhqfVwvRhPXhgYActpYq4KRU5EREXYBgGy3eeZubHB6isdhAV4svC5AQS2jUzO5qImEhFTkTEyZVWVPP0h/tZlXkWgJ91D+eV0bE0C/A2OZmImE1FTkTEiR3JK+aRd3bz9YVSPKwWnhjSjYdv7IjVqqVUEVGRExFxWu/tPsPTH+6jvMpBRLAPC8clcF2HMLNjiYgTUZETEXEyVyrtPPPRflbuPgPAjV1aMO/+OJoH+picTEScjYqciIgT+fpCCRPeySA7rxirBSYndSXl1s5aShWR76QiJyLiJD7KOsv0D/ZRVmmnRaAP88fGcUOnFmbHEhEnpiInImKy8io7z605SNr2HAAGdGzOX8bGER7ka3IyEXF2KnIiIiY6ebGUCUszOHjehsUCE2/tzGNJXfHQUqqI1IGKnIiISf657zxPvreX4opqwgK8mXd/HDd1bWl2LBFxISpyIiKNrKLazux/Hubtr04CcF2HZiwYm0BkiJZSReTqqMiJiDSi0wVlpKZlsOdMEQC/v7kTjw/uiqeH1eRkIuKKVORERBrJvw7k8vjKPdjKqwn19+LV+2L5WfcIs2OJiAtTkRMRaWBVdgcvrT3M37acACC+XSgLxyXQOtTP5GQi4upU5EREGtC5wiukpmWQkVMIwG8HRfOHod3x9tRSqohcu2sqclVVVeTm5lJWVkbLli0JC9NnAIqI/Nvnh/OZ/G4WhWVVBPl68vLoWIb0ijQ7log0IVdd5IqLi3nnnXdYvnw5O3bsoLKyEsMwsFgstGnThsGDB/Pwww9z3XXXNUReERGnV2138Mr6I7z+xdcA9G4dwqJxCbRr7m9yMhFpaq7q3P6rr75Khw4deOutt0hKSuLDDz8kKyuLI0eOkJ6ezsyZM6murmbw4MEMHTqUo0ePNlRuERGnlGcrZ9yb22tK3PgB7XnvkQEqcSLSICyGYRh1HTx27FiefvppevXq9YPjysvLefvtt/H29uaBBx645pDOwmazERISQlFREcHBwWbHEREn8+XRC0xansWl0koCfTx5aVQfhvdpZXYsEWnCrqrI/acbbriBdevWuVWhUZETke9idxj8ZcNRFmw8imFAj1bBLE5OILpFgNnRRKSJ+8mXTW3bto3y8vL/2m6z2XjyySevKZSIiKvILy7nl3/fzvwN35S4sf3bsWrCDSpxItIorrrI3XvvvcyZMweLxUJ+fv5/7S8tLeXll1+ul3AiIs4s/etLDJ+/ha++voS/twfz7o9j9sje+Hp5mB1NRNzEVV+12q5dO9asWYNhGMTGxtK8eXNiY2OJjY0lLi6O7OxsWrXSe0JEpOlyOAwWf3GMV9cfwWFA14hAFif3pXN4oNnRRMTNXHWRe/XVVwHw9vZm69atnDt3jszMTLKysli1ahUOh4O5c+fWe1AREWdwqaSCye/uYfORCwCM7tuG5+6Kwc9bZ+FEpPH95Isdqqqq8PLyqu88Tk0XO4i4t50nC5iYlkmurRxfLyvP3xXD6H5tzY4lIm7sJ3+yg7uVOBFxXw6HwZtfHmfup9nYHQadWgawOLkv3SKDzI4mIm7uqi52yMnJuapvfvbs2asaLyLibArLKnnof3Yxe+1h7A6Du+Oi+Dh1kEqciDiFqypy1113Hb/73e/YuXPn944pKirizTffJCYmhvfff/+aA4qImCUz5zLD529hw+F8vD2tzB7Zm9fujyPA55o+plpEpN5c1W+j4cOHExgYyM9//nN8fX3p27cvUVFR+Pr6cvnyZQ4ePMiBAwdISEhg7ty5DBs2rKFyi4g0GMMw+H9bTzJn7SGq7AYdmvuzKDmBXlEhZkcTEanlqi528Pb25vTp0wQFBdGyZUvGjh3LpUuXuHLlCi1atCA+Pp4hQ4YQExPTkJlNo4sdRJq+oitV/OG9PXx6IA+A4b1bMWdUb4J89b5gEXE+V3VGLioqiqysLIYMGcKVK1d48cUXCQ8Pb6hsIiKNav/ZIiYszSCnoAwvDwtPD+/Jrwa0x2KxmB1NROQ7XVWRmzp1KiNGjCAxMRGLxcLSpUsZNGgQMTEx+Pn5NVRGEZEGZRgG72w7xfNrDlFpd9CmmR+LkxPo0ybU7GgiIj/oqu8jt3fvXlavXs2MGTPo2LEjJ0+exGKx0Llz55pPd4iNjeX2229vqMym0dKqSNNTXF7F9A/2sWbveQB+3jOCl++NJcRfS6ki4vx+8g2Bu3TpQnp6OgEBAezdu5esrKyax/79+ykuLq7vrKZTkRNpWg6es5GSlsGJi6V4Wi1Mu707Dw6K1lKqiLiMn1zkfohhGE3yF6GKnEjTYBgGK3aeZubHB6iodhAV4suCcQn0bd/M7GgiIlelQW6G1BRLnIg0DWWV1Ty9aj8fZH5zw/Jbu7Xk1fviaBbgbXIyEZGrp7taiojbOJJXzISlGRzLL8HDauHxwd343U0dsVr1H58i4ppU5ETELby3+wwzPtzPlSo7EcE+LBibQP/oMLNjiYhcExU5EWnSrlTamfnxft7ddQaAG7u04LX742gR6GNyMhGRa6ciJyJN1tcXSpjwTgbZecVYLTApqSspt3bGQ0upItJEqMiJSJP0UdZZpn+wj7JKOy0CfZg/Jo4bOrcwO5aISL1SkRORJqW8ys5zaw6Stj0HgOs7hjF/bDzhQb4mJxMRqX8qciLSZJy8WMqEpRkcPG/DYoHUWzszKamrllJFpMlSkRORJuGf+87zh/f2UlJRTViAN/Puj+Omri3NjiUi0qBU5ETEpVVU23nxk0P8I/0UANd1aMaCsQlEhmgpVUSaPhU5EXFZpwvKSEnLYO+ZIgB+f3MnHh/cFU8Pq8nJREQah4qciLikfx3I5fGVe7CVVxPq78Wr98Xys+4RZscSEWlUKnIi4lKq7A5eWnuYv205AUB8u1AWjkugdaifyclERBqfipyIuIxzhVdITcsgI6cQgAcHRfPk0O54e2opVUTck4qciLiEzw/nM/ndLArLqgjy9eTl0bEM6RVpdiwREVOpyImIU6u2O3hl/RFe/+JrAHq3DmHRuATaNfc3OZmIiPlU5ETEaeXZypm4LJMdJwoA+NWA9jw1vAc+nh4mJxMRcQ4qciLilL48eoFJy7O4VFpJoI8nc0b15o4+UWbHEhFxKi75DuE5c+ZgsViYNGlSzbby8nJSUlJo3rw5gYGBjBo1iry8vFpfl5OTw/Dhw/H39yc8PJwnnniC6urqRk4vIj/E7jB4bf0RfvX/dnCptJIerYJZPXGQSpyIyHdwuTNyO3fu5K9//St9+vSptX3y5Ml88sknrFy5kpCQEFJTUxk5ciRbt24FwG63M3z4cCIjI/nqq684f/48v/rVr/Dy8uLFF18041BE5FsuFFcwaUUmW49dAmBs/3bMHNETXy8tpYqIfBeLYRiG2SHqqqSkhISEBBYvXsysWbOIi4tj3rx5FBUV0bJlS9LS0rj33nsBOHz4MD169CA9PZ3rr7+etWvXcscdd3Du3DkiIr65aeiSJUt48sknuXDhAt7e3j/6/DabjZCQEIqKiggODm7QYxVxN+lfX+LR5ZlcKK7Az8uDF0fGcE98G7NjiYg4NZdaWk1JSWH48OEkJSXV2r57926qqqpqbe/evTvt2rUjPT0dgPT0dHr37l1T4gCGDBmCzWbjwIED3/l8FRUV2Gy2Wg8RqV8Oh8Giz4+R/LdtXCiuoGtEIKsnDlSJExGpA5dZWl2+fDkZGRns3Lnzv/bl5ubi7e1NaGhore0RERHk5ubWjPnPEvfv/f/e911mz57Ns88+Ww/pReS7FJRWMnlFFpuOXADg3r5teO6uXvh7u8yvJhERU7nEGbnTp0/z2GOPsXTpUnx9fRvteadPn05RUVHN4/Tp04323CJN3a6TBQz7y5dsOnIBXy8rc+/tw8ujY1XiRESugkv8xty9ezf5+fkkJCTUbLPb7WzevJmFCxfy6aefUllZSWFhYa2zcnl5eURGfnPn98jISHbs2FHr+/77qtZ/j/k2Hx8ffHx86vloRNybw2Hw5pfHmftpNnaHQceWASxOTqB7pN53KiJytVzijNxtt93Gvn37yMrKqnn069eP5OTkmv/v5eXFhg0bar4mOzubnJwcBgwYAMCAAQPYt28f+fn5NWPWr19PcHAwPXv2bPRjEnFHhWWVPPQ/u5i99jB2h8FdcVF8nDpIJU5E5CdyiTNyQUFBxMTE1NoWEBBA8+bNa7Y/+OCDTJkyhbCwMIKDg5k4cSIDBgzg+uuvB2Dw4MH07NmTX/7yl8ydO5fc3FyefvppUlJSdNZNpBFk5lwmNS2Ts4VX8Pa0MnNET8b1b4fFYjE7moiIy3KJIlcXr732GlarlVGjRlFRUcGQIUNYvHhxzX4PDw/WrFnDI488woABAwgICGD8+PE899xzJqYWafoMw+CtrSeZvfYQVXaD9s39WTQugZjWIWZHExFxeS51Hzmz6T5yIlen6EoVT763l3UHvrkyfFjvSOaM6kOwr5fJyUREmoYmc0ZORJzL/rNFTFiaQU5BGV4eFp4a1oPxN3TQUqqISD1SkRORemUYBu9sz+H51QeptDto08yPReMSiG0banY0EZEmR0VOROpNSUU1097fy5q95wFI6hHBK6NjCfHXUqqISENQkRORenHovI2UpRkcv1iKp9XCk0O789sbo7WUKiLSgFTkROSaGIbBu7tO88xHB6iodtAqxJeF4xLo276Z2dFERJo8FTkR+cnKKqt5etV+Psg8C8At3Vry6n1xhAV4m5xMRMQ9qMiJyE9yNK+YCUszOJpfgtUCjw/pxu9v6oTVqqVUEZHGoiInIlftg4wzPLVqP1eq7IQH+bBgbDyJHZubHUtExO2oyIlInZVX2Zn50QFW7DoNwKDOLXjt/jhaBulj7kREzKAiJyJ1cvxCCROWZnA4txiLBSbd1pXUn3XGQ0upIiKmUZETkR/18Z5zTH9/L6WVdloEevOXMfEM7NzC7FgiIm5PRU5Evld5lZ1ZnxzknW05ACRGh7FgbDzhwb4mJxMREVCRE5HvcepSKROWZnDgnA2A1Fs7MympC54eVpOTiYjIv6nIich/WbvvPH94by/FFdU08/fitfvjuKVbuNmxRETkW1TkRKRGZbWDF/95iLe/OglAv/bNWDAunlYhfuYGExGR76QiJyIAnC4oIzUtgz1nigD43c0deXxwN7y0lCoi4rRU5ESE9QfzmPpuFrbyakL8vHj1vlhu6xFhdiwREfkRKnIibqzK7uDPn2bzxubjAMS2DWXRuHjaNPM3OZmIiNSFipyImzpXeIWJyzLZfeoyAA8MjGba7d3x9tRSqoiIq1CRE3FDn2fnM2VFFpfLqgjy9eTP9/ZhaEwrs2OJiMhVUpETcSPVdgevrj/C4i++BiCmdTCLx/WlXXMtpYqIuCIVORE3kWcrZ+KyTHacKADgl9e356nhPfD18jA5mYiI/FQqciJuYMvRizy2PJNLpZUEeHswZ1QfRsRGmR1LRESukYqcSBNmdxjM33CU+RuPYhjQPTKIxckJdGwZaHY0ERGpBypyIk3UheIKJq3IZOuxSwCMua4tf7qzl5ZSRUSaEBU5kSZo2/FLTFyWyYXiCvy8PHjhnhhGJrQxO5aIiNQzFTmRJsThMHh909e88q9sHAZ0CQ9kcXICXSKCzI4mIiINQEVOpIkoKK1k8oosNh25AMDIhNbMujsGf2/9mIuINFX6DS/SBOw+VUBqWibni8rx8bTy/F0xjO7XBovFYnY0ERFpQCpyIi7MMAze/PI4c9dlU+0w6NgigEXJCfRoFWx2NBERaQQqciIuqqisiqkr9/DZoTwA7oyN4sWRvQn00Y+1iIi70G98EReUdbqQlKUZnC28grenlZkjejKufzstpYqIuBkVOREXYhgGb391khf/eYgqu0H75v4sGpdATOsQs6OJiIgJVOREXIStvIon39vL2v25ANweE8lL9/Yh2NfL5GQiImIWFTkRF7D/bBETlmaQU1CGl4eFp4b1YPwNHbSUKiLi5lTkRJyYYRgs3Z7Dc6sPUml30DrUj0XJCcS1DTU7moiIOAEVOREnVVJRzfQP9rF6zzkAknqE8/LoWEL9vU1OJiIizkJFTsQJHTpvI2VpBscvluJhtTBtaHd+e2O0llJFRKQWFTkRJ2IYBit3nWHGR/upqHYQGezLwnHx9OsQZnY0ERFxQipyIk6irLKapz/czwcZZwG4uWtLXrs/jrAALaWKiMh3U5ETcQJH84qZsDSDo/klWC0wdXA3Hrm5E1arllJFROT7qciJmOyDjDM8tWo/V6rshAf5MH9sPNd3bG52LBERcQEqciImKa+y86ePD7B852kABnZuzrz742kZ5GNyMhERcRUqciImOH6hhAlLMzicW4zFAo/d1oWJP+uCh5ZSRUTkKqjIiTSy1XvOMe39vZRW2mkR6M28++MZ1KWF2bFERMQFqciJNJLyKjuzPjnIO9tyAEiMDmP+2Hgign1NTiYiIq5KRU6kEZy6VEpKWgb7z9oAmHBLJ6b8vCueHlaTk4mIiCtTkRNpYOv2n+eJlXsprqimmb8Xr94fx63dws2OJSIiTYCKnEgDqax2MHvtId7aehKAvu2bsWBsPFGhfuYGExGRJkNFTqQBnLlcRkpaJntOFwLw8E0deWJIN7y0lCoiIvVIRU6knq0/mMfUd7OwlVcT4ufFK6NjSeoZYXYsERFpglTkROpJld3By59m89fNxwGIbRvKwrHxtA3zNzmZiIg0VSpyIvXgfNEVUtMy2X3qMgC/GdiB6bf3wNtTS6kiItJwVORErtEX2flMXpHF5bIqgnw8mXtvH27v3crsWCIi4gZU5ER+omq7g3mfHWXh58cAiGkdzKJxCbRvHmByMhERcRcqciI/Qb6tnInLMtl+ogCAX1zfjqeH98TXy8PkZCIi4k5U5ESu0lfHLvLo8kwullQS4O3B7FF9uDM2yuxYIiLihlTkROrI7jBYuPEY8zYcwTCge2QQi5IT6NQy0OxoIiLiplTkROrgYkkFk5ZnseXYRQDGXNeWP93ZS0upIiJiKhU5kR+x/fglJi7LJL+4Aj8vD2bdHcOovm3MjiUiIqIiJ/J9HA6DJZu/5uVPs3EY0Dk8kNeTE+gSEWR2NBEREUBFTuQ7XS6tZMq7WXyefQGAkfGtmXVPDP7e+pERERHnoVclkW/ZfaqA1LRMzheV4+Np5bm7enFfv7ZYLBazo4mIiNSiIifyfwzD4G9fnuCldYepdhhEtwhgcXICPVoFmx1NRETkO6nIiQBFZVVMXbmHzw7lATAiNorZI3sT6KMfERERcV56lRK3t+d0ISlpGZy5fAVvDyszRvTkF4nttJQqIiJOT0VO3JZhGPzjq5O88M9DVNkN2oX5szg5gZjWIWZHExERqRMVOXFLtvIqpr2/l3/uywVgSK8I5t4bS4ifl8nJRERE6s5qdoC6mD17Ntdddx1BQUGEh4dz9913k52dXWtMeXk5KSkpNG/enMDAQEaNGkVeXl6tMTk5OQwfPhx/f3/Cw8N54oknqK6ubsxDESew/2wRIxZs4Z/7cvHysDDjjp4s+UVflTgREXE5LlHkNm3aREpKCtu2bWP9+vVUVVUxePBgSktLa8ZMnjyZ1atXs3LlSjZt2sS5c+cYOXJkzX673c7w4cOprKzkq6++4h//+Advv/02zzzzjBmHJCYwDIN3tp1i5OtfcepSGa1D/Vj5+xt4cFC03g8nIiIuyWIYhmF2iKt14cIFwsPD2bRpEzfddBNFRUW0bNmStLQ07r33XgAOHz5Mjx49SE9P5/rrr2ft2rXccccdnDt3joiICACWLFnCk08+yYULF/D29v7R57XZbISEhFBUVERwsG5J4UpKKqr54wf7+HjPOQBu6x7OK/fFEur/4/MuIiLirFzijNy3FRUVARAWFgbA7t27qaqqIikpqWZM9+7dadeuHenp6QCkp6fTu3fvmhIHMGTIEGw2GwcOHPjO56moqMBms9V6iOs5nGvjzgVb+HjPOTysFqbf3p03f9VPJU5ERFyeyxU5h8PBpEmTGDhwIDExMQDk5ubi7e1NaGhorbERERHk5ubWjPnPEvfv/f/e911mz55NSEhIzaNt27b1fDTSkAzD4N2dp7lr4VaOXywlMtiXFQ9fz+9u7oTVqqVUERFxfS5X5FJSUti/fz/Lly9v8OeaPn06RUVFNY/Tp083+HNK/SirrGbqyj384f29VFQ7uKlrSz55dBD9OoSZHU1ERKTeuNTtR1JTU1mzZg2bN2+mTZs2NdsjIyOprKyksLCw1lm5vLw8IiMja8bs2LGj1vf791Wt/x7zbT4+Pvj4+NTzUUhDO5ZfzCPvZHA0vwSrBaYO7sYjOgsnIiJNkEuckTMMg9TUVFatWsXGjRuJjo6utb9v3754eXmxYcOGmm3Z2dnk5OQwYMAAAAYMGMC+ffvIz8+vGbN+/XqCg4Pp2bNn4xyINLhVmWcYsWArR/NLaBnkw9LfXk/KrZ1V4kREpElyiatWJ0yYQFpaGh999BHdunWr2R4SEoKfnx8AjzzyCP/85z95++23CQ4OZuLEiQB89dVXwDe3H4mLiyMqKoq5c+eSm5vLL3/5S37729/y4osv1imHrlp1XuVVdv708QGW7/xm+fuGTs35y5h4WgbpjKqIiDRdLlHkvu8eX2+99Ra//vWvgW9uCDx16lSWLVtGRUUFQ4YMYfHixbWWTU+dOsUjjzzCF198QUBAAOPHj2fOnDl4etZthVlFzjkdv1DChKUZHM4txmKBR3/WhUdv64KHzsKJiEgT5xJFzlmoyDmf1XvOMe39vZRW2mke4M28MXHc2KWl2bFEREQahUtd7CDyb+VVdmZ9cpB3tuUA0D86jAVj44kI9jU5mYiISONRkROXk3OpjAlpu9l/9psbNE+4pRNTft4VTw+XuHZHRESk3qjIiUtZtz+XJ97bQ3F5NaH+Xrx2fxy3dgs3O5aIiIgpVOTEJVRWO5i99hBvbT0JQEK7UBaOSyAq1M/cYCIiIiZSkROnd+ZyGSlpmew5XQjAQzdG84eh3fHSUqqIiLg5FTlxahsO5THl3T0UXaki2NeTl0fHMrjXd38Sh4iIiLtRkROnVGV38PKn2fx183EA+rQJYdG4BNqG+ZucTERExHmoyInTOV90hYlpmew6dRmAX9/QgenDuuPj6WFyMhEREeeiIidOZdORC0xekUVBaSVBPp68dG8fhvVuZXYsERERp6QiJ07B7jCY99kRFn5+DMOAnq2CWZycQIcWAWZHExERcVoqcmK6fFs5jy7PZNvxAgDGJbbjmTt64uulpVQREZEfoiInpvrq2EUeXZ7FxZIK/L09mD2yN3fFtTY7loiIiEtQkRNT2B0GCzceY96GIxgGdIsIYlFyAp3DA82OJiIi4jJU5KTRXSypYNLyLLYcuwjAff3a8OydMfh5aylVRETkaqjISaPafvwSE5dlkl9cga+XlVl39+bevm3MjiUiIuKSVOSkUTgcBks2f83Ln2bjMKBzeCCLkxPoGhFkdjQRERGXpSInDe5yaSVT3s3i8+wLANwT35pZd8cQ4KN/fiIiItdCr6TSoHafKiA1LZPzReV4e1p57s5e3H9dWywWi9nRREREXJ6KnDQIwzD4+5YTzFl7mGqHQXSLABaNS6BnVLDZ0URERJoMFTmpd0VlVTz+3h7WH8wDYHifVswZ2ZsgXy+Tk4mIiDQtKnJSr/aeKWTC0gzOXL6Ct4eVGXf04BfXt9dSqoiISANQkZN6YRgG/5N+ihc+OUSl3UHbMD8Wj+tL7zYhZkcTERFpslTk5JrZyquY/v4+Ptl3HoAhvSKYe28sIX5aShUREWlIKnJyTQ6cKyJlaQYnL5XhabUwfVgPHhjYQUupIiIijUBFTn4SwzBYvvM0Mz8+QGW1g9ahfiwcF098u2ZmRxMREXEbKnJy1Uorqnlq1T4+zDoHwG3dw3nlvlhC/b1NTiYiIuJeVOTkqmTnFjNh6W6+vlCKh9XCE0O68fCNHbFatZQqIiLS2FTkpM5W7jrNjI/2U17lIDLYl4Xj4unXIczsWCIiIm5LRU5+1JVKOzM+2s97u88AcFPXlrx2XyzNA31MTiYiIuLeVOTkBx3LLyFlaQbZecVYLTB1cDceubmTllJFREScgIqcfK+Pss4y/YN9lFXaaRnkw/wx8Qzo1NzsWCIiIvJ/VOTkv5RX2Xl29UGW7cgB4IZOzfnLmHhaBmkpVURExJmoyEktJy+WMmFpBgfP27BY4NGfdeHR27rgoaVUERERp6MiJzU+2XueJ9/fS0lFNc0DvJk3Jo4bu7Q0O5aIiIh8DxU5oaLazoufHOIf6acA6B8dxoKx8UQE+5qcTERERH6IipybO11QRkpaBnvPFAEw4ZZOTPl5Vzw9rCYnExERkR+jIufG/nUgl8dX7sFWXk2ovxev3RfHrd3DzY4lIiIidaQi54aq7A5eWnuYv205AUBCu1AWjksgKtTP5GQiIiJyNVTk3MzZwiukpmWQmVMIwEM3RvOHod3x0lKqiIiIy1GRcyMbD+cx5d09FJZVEezrycujYxncK9LsWCIiIvITqci5gWq7g5f/dYQlm74GILZNCAvHJdA2zN/kZCIiInItVOSauNyich5dlsmOkwUA/PqGDkwf1h0fTw+Tk4mIiMi1UpFrwjYfucCkFVkUlFYS6OPJS6P6MLxPK7NjiYiISD1RkWuC7A6DeZ8dYeHnxzAM6NkqmMXJCXRoEWB2NBEREalHKnJNTH5xOY8tyyL9+CUAxiW245k7euLrpaVUERGRpkZFrgn56thFHl2excWSCvy9PZg9sjd3xbU2O5aIiIg0EBW5JsDuMFj0+THmfXYEhwHdIoJYlJxA5/BAs6OJiIhIA1KRc3EXSyqYvCKLL49eBGB03zY8d1cMft5aShUREWnqVORc2Pbjl3h0eSZ5tgp8vaw8f1cMo/u1NTuWiIiINBIVORfkcBi8vulrXvlXNg4DOrUMYHFyX7pFBpkdTURERBqRipyLKSitZMq7WXyRfQGAe+JbM+vuGAJ8NJUiIiLuRq/+LmTXyQImLsvkfFE5Pp5Wnr2zF/df1xaLxWJ2NBERETGBipwLcDgM/rblOC+ty8buMOjYIoBFyQn0aBVsdjQRERExkYqckyssq2Tqu3vYcDgfgBGxUcwe2ZtALaWKiIi4PbUBJ5aRc5mJaZmcLbyCt6eVmSN6Mq5/Oy2lioiICKAi55QMw+DvW04wZ+1hqh0GHZr7s3BcAjGtQ8yOJiIiIk5ERc7JFJVVMXXlHj47lAfA8D6tmDOyN0G+XiYnExEREWejIudEsk4XkrI045ulVA8rM0b05BeJWkoVERGR76Yi5wQcjm+WUl9a981Savvm/izSUqqIiIj8CBU5kxWUVvL4yj1s/L+rUof3bsXsUb0J1lKqiIiI/AgVORPtOFHAo8syybWV66pUERERuWoqciawOwwWf36M1z47gsOAji0DWDRON/gVERGRq6Mi18hyi8qZujKLrccuATAqoQ3P3dVLn5UqIiIiV03toRGt25/LtA/2UlhWhb+3B8/fFcOovm3MjiUiIiIuSkWuEZRVVvP8moMs23EagN6tQ5g3Jo5OLQNNTiYiIiKuTEWuge07U8RjyzM5frEUiwV+d1Mnpvy8K96eVrOjiYiIiItTkWsgVyrtzN94lDc3H6faYRAZ7Mur98dyQ6cWZkcTERGRJkJFrgFsPJzHMx8d4MzlKwAM6x3Ji/f0JtTf2+RkIiIi0pSoyNWjc4VXeHb1AT498M3npLYO9eNPd/bi5z0jTE4mIiIiTZFbvlFr0aJFdOjQAV9fXxITE9mxY8c1fb9qu4O/fXmcpFc38emBPDytFn53c0fWT7lJJU5EREQajNudkVuxYgVTpkxhyZIlJCYmMm/ePIYMGUJ2djbh4eFX9b1s5VWs3HWG/0k/yalLZQD0bd+MF+6JoXukbu4rIiIiDctiGIZhdojGlJiYyHXXXcfChQsBcDgctG3blokTJzJt2rQf/NqtB3MY1Ks9r/9rLwcvVrPhUB6llXYAQv29mDa0O/f1a4vVqo/YEhERkYbXZM7IGYZBcXHxD46prKxk165dPPbYY9hstprtN910E5s3b2bChAm1xldUVFBRUVHz5zfW7wFg1qoMrD7+AHRqGcC4xHbc0SeKAB9PSkp+OIOIiIhIXQUFBf3gZ7A3mTNyNpuNkJAQs2OIiIiI1JuioiKCg7//7VpNpsjV5Yzc+fPn6d69O+vXr6d///4122fMmMHWrVvZuHFjrfHfPiN3/vx5+vfvz8GDB2ndunX9HoBcE5vNRtu2bTl9+vQP/oOXxqe5cV6aG+eluXFejT03P3ZGrsksrVoslh/9C/X19cXDw4OSkpJaYwsLC2ndunWdJyQoKEg/WE4qODhYc+OkNDfOS3PjvDQ3zstZ5satbj/i7e1N37592bBhQ802h8PBhg0bGDBggInJRERERK5ekzkjV1dTpkxh/Pjx9OvXj/79+zNv3jxKS0v5zW9+Y3Y0ERERkavidkXu/vvv58KFCzzzzDPk5uYSFxfHunXriIj48Rv3+vj41PpfcR4+Pj7MnDlTc+OENDfOS3PjvDQ3zsvZ5qbJXOzQGP59ZeyPXUEiIiIi0hjc6j1yIiIiIk2JipyIiIiIi1KRExEREXFRKnIiIiIiLkpF7iq88cYbALRs2ZLExER27NhhciL3M3v2bK677jqCgoIIDw/n7rvvJjs7u9aY8vJyUlJSaN68OYGBgYwaNYq8vDyTErunOXPmYLFYmDRpUs02zYt5zp49yy9+8QuaN2+On58fvXv3ZteuXTX7DcPgmWeeoVWrVvj5+ZGUlMTRo0dNTOwe7HY7M2bMIDo6Gj8/Pzp16sTzzz/Pf16DqLlpHJs3b2bEiBFERUVhsVj48MMPa+2vyzwUFBSQnJxMcHAwoaGhPPjgg5SUlDR4dhW5OlqxYgV//OMfAfjyyy+JjY1lyJAh5Ofnm5zMvWzatImUlBS2bdvG+vXrqaqqYvDgwZSWltaMmTx5MqtXr2blypVs2rSJc+fOMXLkSBNTu5edO3fy17/+lT59+tTarnkxx+XLlxk4cCBeXl6sXbuWgwcP8sorr9CsWbOaMXPnzmX+/PksWbKE7du3ExAQwJAhQygvLzcxedP30ksv8frrr7Nw4UIOHTrESy+9xNy5c1mwYEHNGM1N4ygtLSU2NpZFixZ95/66zENycjIHDhxg/fr1rFmzhs2bN/Pwww83fHhD6qR///7GQw89ZABGUVGRYbfbjaioKGP27NlmR3Nr+fn5BmBs2rTJMAzDKCwsNLy8vIyVK1fWjDl06JABGOnp6WbFdBvFxcVGly5djPXr1xs333yz8dhjjxmGoXkx05NPPmkMGjToe/c7HA4jMjLS+POf/1yzrbCw0PDx8TGWLVvWGBHd1vDhw40HHnig1raRI0caycnJhmFobswCGKtWrar5c13m4eDBgwZg7Ny5s2bM2rVrDYvFYpw9e7ZB8+qMXB1UVlaye/dubrnllpptVquVpKQk0tPTzQsmFBUVARAWFgbA7t27qaqqIikpqWZM9+7dadeuneaqEaSkpDB8+PBaf/+geTHTxx9/TL9+/Rg9ejTh4eHEx8fz5ptv1uw/ceIEubm5teYmJCSExMREzU0Du+GGG9iwYQNHjhwBYM+ePWzZsoXbb78d0Nw4i7rMQ3p6OqGhofTr169mTFJSElarle3btzdoPrf7ZIef4uLFi9jtdsLDw2ttj4iI4PDhwyalEofDwaRJkxg4cCAxMTEA5Obm4u3tTWhoaK2xERER5ObmmpDSfSxfvpyMjAx27tz5X/s0L+Y5fvw4r7/+OlOmTOGPf/wjO3fu5NFHH8Xb25vx48fX/P1/+9NtNDcNb9q0adhsNrp3746Hhwd2u50XXniB5ORkAM2Nk6jLPOTm5v5XR/D09CQsLKzB50pFTlxWSkoK+/fvZ8uWLWZHcXunT5/mscceY/369fj6+podR/6Dw+GgX79+vPjiiwDEx8ezf/9+lixZwvjx401O597effddli5dSlpaGr169SIrK4tJkyYRFRWluZE609JqHbRo0QIPD4//urAhLy+PyMhIk1K5t9TUVNasWcPnn39OmzZtarZHRkZSWVlJYWFhrfGaq4a1e/du8vPzSUhIwNPTE09PTzZt2sT8+fPx9PQkIiJC82KSVq1a0bNnz1rbevToQU5ODkDN3/+3ryDW3DS8J554gmnTpjFmzBh69+7NL3/5SyZPnszs2bMBzY2zqMs8REZG/ldHqK6upqCgoMHnSkWuDry9venbty+bNm2q2eZwONiwYQMDBgwwMZn7MQyD1NRUVq1axcaNG4mOjq61v2/fvnh5ebFhw4aabdnZ2eTk5GiuGtBtt93Gvn37yMrKqnn069eP5OTkmv+veTHHwIED/+sWPUeOHKF9+/YAREdHExkZWWtubDYb27dv19w0sLKyMqzW2i/DHh4eOBwOQHPjLOoyDwMGDKCwsJDdu3fXjNm4cSMOh4PExMSGDdigl1I0IcuXLze8vb0NwNixY4fx8MMPG6GhoUZubq7Z0dzKI488YoSEhBhffPGFcf78+ZpHWVlZzZjf//73Rrt27YyNGzcau3btMgYMGGAMGDDAxNTu6T+vWjUMzYtZduzYYXh6ehovvPCCcfToUWPp0qWGv7+/8c4779SMmTNnjhEaGmp89NFHxt69e4277rrLiI6ONq5cuWJi8qZv/PjxRuvWrY01a9YYJ06cMD744AOjRYsWxh/+8IeaMZqbxlFcXGxkZmYamZmZBmC8+uqrRmZmpnHq1CnDMOo2D0OHDjXi4+ON7du3G1u2bDG6dOlijB07tsGzq8hdhfnz5xtt2rQxvLy8jP79+xvbtm0zO5LbAb7z8dZbb9WMuXLlijFhwgSjWbNmhr+/v3HPPfcY58+fNy+0m/p2kdO8mGf16tVGTEyM4ePjY3Tv3t144403au13OBzGjBkzjIiICMPHx8e47bbbjOzsbJPSug+bzWY89thjRrt27QxfX1+jY8eOxlNPPWVUVFTUjNHcNI7PP//8O19bxo8fbxhG3ebh0qVLxtixY43AwEAjODjY+M1vfmMUFxc3eHaLYfzHLaRFRERExGXoPXIiIiIiLkpFTkRERMRFqciJiIiIuCgVOREREREXpSInIiIi4qJU5ERERERclIqciIiIiItSkRMRERFxUSpyIiIiIi5KRU5E5CpMnjyZkSNHmh1DRARQkRMRuSo7duygX79+ZscQEQFAn7UqIlIHlZWVBAQEUF1dXbMtMTGRbdu2mZhKRNydp9kBRERcgaenJ1u3biUxMZGsrCwiIiLw9fU1O5aIuDkVORGROrBarZw7d47mzZsTGxtrdhwREUDvkRMRqbPMzEyVOBFxKipyIiJ1lJWVpSInIk5FRU5EpI727dtHXFyc2TFERGqoyImI1JHD4SA7O5tz585RVFRkdhwRERU5EZG6mjVrFm+//TatW7dm1qxZZscREdF95ERERERclc7IiYiIiLgoFTkRERERF6UiJyIiIuKiVOREREREXJSKnIiIiIiLUpETERERcVEqciIiIiIuSkVORERExEWpyImIiIi4KBU5ERERERelIiciIiLiov4/KYS7Glv9oYMAAAAASUVORK5CYII=\n"
          },
          "metadata": {}
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<sympy.plotting.plot.Plot at 0x7f994b8bf1c0>"
            ]
          },
          "metadata": {},
          "execution_count": 53
        }
      ]
    }
  ]
}