// SPDX-License-Identifier: MIT

/*
 * ABDK Math 64.64 Smart Contract Library.  Copyright © 2019 by ABDK Consulting.
 * Author: Mikhail Vladimirov <mikhail.vladimirov@gmail.com>
 */
pragma solidity ^0.8.0;

/**
 * Smart contract library of mathematical functions operating with signed
 * 64.64-bit fixed point numbers.  Signed 64.64-bit fixed point number is
 * basically a simple fraction whose numerator is signed 128-bit integer and
 * denominator is 2^64.  As long as denominator is always the same, there is no
 * need to store it, thus in Solidity signed 64.64-bit fixed point numbers are
 * represented by int128 type holding only the numerator.
 */
library ABDKMath64x64 {
    /*
     * Minimum value signed 64.64-bit fixed point number may have.
     */
    int128 private constant MIN_64x64 = -0x80000000000000000000000000000000;

    /*
     * Maximum value signed 64.64-bit fixed point number may have.
     */
    int128 private constant MAX_64x64 = 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF;

    /**
     * Convert signed 256-bit integer number into signed 64.64-bit fixed point
     * number.  Revert on overflow.
     *
     * @param x signed 256-bit integer number
     * @return signed 64.64-bit fixed point number
     */
    function fromInt(int256 x) internal pure returns (int128) {
        unchecked {
            require(x >= -0x8000000000000000 && x <= 0x7FFFFFFFFFFFFFFF);
            return int128(x << 64);
        }
    }

    /**
     * Convert signed 64.64 fixed point number into signed 64-bit integer number
     * rounding down.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64-bit integer number
     */
    function toInt(int128 x) internal pure returns (int64) {
        unchecked {
            return int64(x >> 64);
        }
    }

    /**
     * Convert unsigned 256-bit integer number into signed 64.64-bit fixed point
     * number.  Revert on overflow.
     *
     * @param x unsigned 256-bit integer number
     * @return signed 64.64-bit fixed point number
     */
    function fromUInt(uint256 x) internal pure returns (int128) {
        unchecked {
            require(x <= 0x7FFFFFFFFFFFFFFF);
            return int128(int256(x << 64));
        }
    }

    /**
     * Convert signed 64.64 fixed point number into unsigned 64-bit integer
     * number rounding down.  Revert on underflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return unsigned 64-bit integer number
     */
    function toUInt(int128 x) internal pure returns (uint64) {
        unchecked {
            require(x >= 0);
            return uint64(uint128(x >> 64));
        }
    }

    /**
     * Convert signed 128.128 fixed point number into signed 64.64-bit fixed point
     * number rounding down.  Revert on overflow.
     *
     * @param x signed 128.128-bin fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function from128x128(int256 x) internal pure returns (int128) {
        unchecked {
            int256 result = x >> 64;
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Convert signed 64.64 fixed point number into signed 128.128 fixed point
     * number.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 128.128 fixed point number
     */
    function to128x128(int128 x) internal pure returns (int256) {
        unchecked {
            return int256(x) << 64;
        }
    }

    /**
     * Calculate x + y.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function add(int128 x, int128 y) internal pure returns (int128) {
        unchecked {
            int256 result = int256(x) + y;
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Calculate x - y.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function sub(int128 x, int128 y) internal pure returns (int128) {
        unchecked {
            int256 result = int256(x) - y;
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Calculate x * y rounding down.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function mul(int128 x, int128 y) internal pure returns (int128) {
        unchecked {
            int256 result = (int256(x) * y) >> 64;
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Calculate x * y rounding towards zero, where x is signed 64.64 fixed point
     * number and y is signed 256-bit integer number.  Revert on overflow.
     *
     * @param x signed 64.64 fixed point number
     * @param y signed 256-bit integer number
     * @return signed 256-bit integer number
     */
    function muli(int128 x, int256 y) internal pure returns (int256) {
        unchecked {
            if (x == MIN_64x64) {
                require(
                    y >= -0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF &&
                        y <= 0x1000000000000000000000000000000000000000000000000
                );
                return -y << 63;
            } else {
                bool negativeResult = false;
                if (x < 0) {
                    x = -x;
                    negativeResult = true;
                }
                if (y < 0) {
                    y = -y; // We rely on overflow behavior here
                    negativeResult = !negativeResult;
                }
                uint256 absoluteResult = mulu(x, uint256(y));
                if (negativeResult) {
                    require(
                        absoluteResult <=
                            0x8000000000000000000000000000000000000000000000000000000000000000
                    );
                    return -int256(absoluteResult); // We rely on overflow behavior here
                } else {
                    require(
                        absoluteResult <=
                            0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
                    );
                    return int256(absoluteResult);
                }
            }
        }
    }

    /**
     * Calculate x * y rounding down, where x is signed 64.64 fixed point number
     * and y is unsigned 256-bit integer number.  Revert on overflow.
     *
     * @param x signed 64.64 fixed point number
     * @param y unsigned 256-bit integer number
     * @return unsigned 256-bit integer number
     */
    function mulu(int128 x, uint256 y) internal pure returns (uint256) {
        unchecked {
            if (y == 0) return 0;

            require(x >= 0);

            uint256 lo = (uint256(int256(x)) *
                (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)) >> 64;
            uint256 hi = uint256(int256(x)) * (y >> 128);

            require(hi <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
            hi <<= 64;

            require(
                hi <=
                    0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF -
                        lo
            );
            return hi + lo;
        }
    }

    /**
     * Calculate x / y rounding towards zero.  Revert on overflow or when y is
     * zero.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function div(int128 x, int128 y) internal pure returns (int128) {
        unchecked {
            require(y != 0);
            int256 result = (int256(x) << 64) / y;
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Calculate x / y rounding towards zero, where x and y are signed 256-bit
     * integer numbers.  Revert on overflow or when y is zero.
     *
     * @param x signed 256-bit integer number
     * @param y signed 256-bit integer number
     * @return signed 64.64-bit fixed point number
     */
    function divi(int256 x, int256 y) internal pure returns (int128) {
        unchecked {
            require(y != 0);

            bool negativeResult = false;
            if (x < 0) {
                x = -x; // We rely on overflow behavior here
                negativeResult = true;
            }
            if (y < 0) {
                y = -y; // We rely on overflow behavior here
                negativeResult = !negativeResult;
            }
            uint128 absoluteResult = divuu(uint256(x), uint256(y));
            if (negativeResult) {
                require(absoluteResult <= 0x80000000000000000000000000000000);
                return -int128(absoluteResult); // We rely on overflow behavior here
            } else {
                require(absoluteResult <= 0x7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
                return int128(absoluteResult); // We rely on overflow behavior here
            }
        }
    }

    /**
     * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
     * integer numbers.  Revert on overflow or when y is zero.
     *
     * @param x unsigned 256-bit integer number
     * @param y unsigned 256-bit integer number
     * @return signed 64.64-bit fixed point number
     */
    function divu(uint256 x, uint256 y) internal pure returns (int128) {
        unchecked {
            require(y != 0);
            uint128 result = divuu(x, y);
            require(result <= uint128(MAX_64x64));
            return int128(result);
        }
    }

    /**
     * Calculate -x.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function neg(int128 x) internal pure returns (int128) {
        unchecked {
            require(x != MIN_64x64);
            return -x;
        }
    }

    /**
     * Calculate |x|.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function abs(int128 x) internal pure returns (int128) {
        unchecked {
            require(x != MIN_64x64);
            return x < 0 ? -x : x;
        }
    }

    /**
     * Calculate 1 / x rounding towards zero.  Revert on overflow or when x is
     * zero.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function inv(int128 x) internal pure returns (int128) {
        unchecked {
            require(x != 0);
            int256 result = int256(0x100000000000000000000000000000000) / x;
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Calculate arithmetics average of x and y, i.e. (x + y) / 2 rounding down.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function avg(int128 x, int128 y) internal pure returns (int128) {
        unchecked {
            return int128((int256(x) + int256(y)) >> 1);
        }
    }

    /**
     * Calculate geometric average of x and y, i.e. sqrt (x * y) rounding down.
     * Revert on overflow or in case x * y is negative.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function gavg(int128 x, int128 y) internal pure returns (int128) {
        unchecked {
            int256 m = int256(x) * int256(y);
            require(m >= 0);
            require(
                m <
                    0x4000000000000000000000000000000000000000000000000000000000000000
            );
            return int128(sqrtu(uint256(m)));
        }
    }

    /**
     * Calculate x^y assuming 0^0 is 1, where x is signed 64.64 fixed point number
     * and y is unsigned 256-bit integer number.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @param y uint256 value
     * @return signed 64.64-bit fixed point number
     */
    function pow(int128 x, uint256 y) internal pure returns (int128) {
        unchecked {
            bool negative = x < 0 && y & 1 == 1;

            uint256 absX = uint128(x < 0 ? -x : x);
            uint256 absResult;
            absResult = 0x100000000000000000000000000000000;

            if (absX <= 0x10000000000000000) {
                absX <<= 63;
                while (y != 0) {
                    if (y & 0x1 != 0) {
                        absResult = (absResult * absX) >> 127;
                    }
                    absX = (absX * absX) >> 127;

                    if (y & 0x2 != 0) {
                        absResult = (absResult * absX) >> 127;
                    }
                    absX = (absX * absX) >> 127;

                    if (y & 0x4 != 0) {
                        absResult = (absResult * absX) >> 127;
                    }
                    absX = (absX * absX) >> 127;

                    if (y & 0x8 != 0) {
                        absResult = (absResult * absX) >> 127;
                    }
                    absX = (absX * absX) >> 127;

                    y >>= 4;
                }

                absResult >>= 64;
            } else {
                uint256 absXShift = 63;
                if (absX < 0x1000000000000000000000000) {
                    absX <<= 32;
                    absXShift -= 32;
                }
                if (absX < 0x10000000000000000000000000000) {
                    absX <<= 16;
                    absXShift -= 16;
                }
                if (absX < 0x1000000000000000000000000000000) {
                    absX <<= 8;
                    absXShift -= 8;
                }
                if (absX < 0x10000000000000000000000000000000) {
                    absX <<= 4;
                    absXShift -= 4;
                }
                if (absX < 0x40000000000000000000000000000000) {
                    absX <<= 2;
                    absXShift -= 2;
                }
                if (absX < 0x80000000000000000000000000000000) {
                    absX <<= 1;
                    absXShift -= 1;
                }

                uint256 resultShift = 0;
                while (y != 0) {
                    require(absXShift < 64);

                    if (y & 0x1 != 0) {
                        absResult = (absResult * absX) >> 127;
                        resultShift += absXShift;
                        if (absResult > 0x100000000000000000000000000000000) {
                            absResult >>= 1;
                            resultShift += 1;
                        }
                    }
                    absX = (absX * absX) >> 127;
                    absXShift <<= 1;
                    if (absX >= 0x100000000000000000000000000000000) {
                        absX >>= 1;
                        absXShift += 1;
                    }

                    y >>= 1;
                }

                require(resultShift < 64);
                absResult >>= 64 - resultShift;
            }
            int256 result = negative ? -int256(absResult) : int256(absResult);
            require(result >= MIN_64x64 && result <= MAX_64x64);
            return int128(result);
        }
    }

    /**
     * Calculate sqrt (x) rounding down.  Revert if x < 0.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function sqrt(int128 x) internal pure returns (int128) {
        unchecked {
            require(x >= 0);
            return int128(sqrtu(uint256(int256(x)) << 64));
        }
    }

    /**
     * Calculate binary logarithm of x.  Revert if x <= 0.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function log_2(int128 x) internal pure returns (int128) {
        unchecked {
            require(x > 0);

            int256 msb = 0;
            int256 xc = x;
            if (xc >= 0x10000000000000000) {
                xc >>= 64;
                msb += 64;
            }
            if (xc >= 0x100000000) {
                xc >>= 32;
                msb += 32;
            }
            if (xc >= 0x10000) {
                xc >>= 16;
                msb += 16;
            }
            if (xc >= 0x100) {
                xc >>= 8;
                msb += 8;
            }
            if (xc >= 0x10) {
                xc >>= 4;
                msb += 4;
            }
            if (xc >= 0x4) {
                xc >>= 2;
                msb += 2;
            }
            if (xc >= 0x2) msb += 1; // No need to shift xc anymore

            int256 result = (msb - 64) << 64;
            uint256 ux = uint256(int256(x)) << uint256(127 - msb);
            for (int256 bit = 0x8000000000000000; bit > 0; bit >>= 1) {
                ux *= ux;
                uint256 b = ux >> 255;
                ux >>= 127 + b;
                result += bit * int256(b);
            }

            return int128(result);
        }
    }

    /**
     * Calculate natural logarithm of x.  Revert if x <= 0.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function ln(int128 x) internal pure returns (int128) {
        unchecked {
            require(x > 0);

            return
                int128(
                    int256(
                        (uint256(int256(log_2(x))) *
                            0xB17217F7D1CF79ABC9E3B39803F2F6AF) >> 128
                    )
                );
        }
    }

    /**
     * Calculate binary exponent of x.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function exp_2(int128 x) internal pure returns (int128) {
        unchecked {
            require(x < 0x400000000000000000); // Overflow

            if (x < -0x400000000000000000) return 0; // Underflow

            uint256 result = 0x80000000000000000000000000000000;

            if (x & 0x8000000000000000 > 0)
                result = (result * 0x16A09E667F3BCC908B2FB1366EA957D3E) >> 128;
            if (x & 0x4000000000000000 > 0)
                result = (result * 0x1306FE0A31B7152DE8D5A46305C85EDEC) >> 128;
            if (x & 0x2000000000000000 > 0)
                result = (result * 0x1172B83C7D517ADCDF7C8C50EB14A791F) >> 128;
            if (x & 0x1000000000000000 > 0)
                result = (result * 0x10B5586CF9890F6298B92B71842A98363) >> 128;
            if (x & 0x800000000000000 > 0)
                result = (result * 0x1059B0D31585743AE7C548EB68CA417FD) >> 128;
            if (x & 0x400000000000000 > 0)
                result = (result * 0x102C9A3E778060EE6F7CACA4F7A29BDE8) >> 128;
            if (x & 0x200000000000000 > 0)
                result = (result * 0x10163DA9FB33356D84A66AE336DCDFA3F) >> 128;
            if (x & 0x100000000000000 > 0)
                result = (result * 0x100B1AFA5ABCBED6129AB13EC11DC9543) >> 128;
            if (x & 0x80000000000000 > 0)
                result = (result * 0x10058C86DA1C09EA1FF19D294CF2F679B) >> 128;
            if (x & 0x40000000000000 > 0)
                result = (result * 0x1002C605E2E8CEC506D21BFC89A23A00F) >> 128;
            if (x & 0x20000000000000 > 0)
                result = (result * 0x100162F3904051FA128BCA9C55C31E5DF) >> 128;
            if (x & 0x10000000000000 > 0)
                result = (result * 0x1000B175EFFDC76BA38E31671CA939725) >> 128;
            if (x & 0x8000000000000 > 0)
                result = (result * 0x100058BA01FB9F96D6CACD4B180917C3D) >> 128;
            if (x & 0x4000000000000 > 0)
                result = (result * 0x10002C5CC37DA9491D0985C348C68E7B3) >> 128;
            if (x & 0x2000000000000 > 0)
                result = (result * 0x1000162E525EE054754457D5995292026) >> 128;
            if (x & 0x1000000000000 > 0)
                result = (result * 0x10000B17255775C040618BF4A4ADE83FC) >> 128;
            if (x & 0x800000000000 > 0)
                result = (result * 0x1000058B91B5BC9AE2EED81E9B7D4CFAB) >> 128;
            if (x & 0x400000000000 > 0)
                result = (result * 0x100002C5C89D5EC6CA4D7C8ACC017B7C9) >> 128;
            if (x & 0x200000000000 > 0)
                result = (result * 0x10000162E43F4F831060E02D839A9D16D) >> 128;
            if (x & 0x100000000000 > 0)
                result = (result * 0x100000B1721BCFC99D9F890EA06911763) >> 128;
            if (x & 0x80000000000 > 0)
                result = (result * 0x10000058B90CF1E6D97F9CA14DBCC1628) >> 128;
            if (x & 0x40000000000 > 0)
                result = (result * 0x1000002C5C863B73F016468F6BAC5CA2B) >> 128;
            if (x & 0x20000000000 > 0)
                result = (result * 0x100000162E430E5A18F6119E3C02282A5) >> 128;
            if (x & 0x10000000000 > 0)
                result = (result * 0x1000000B1721835514B86E6D96EFD1BFE) >> 128;
            if (x & 0x8000000000 > 0)
                result = (result * 0x100000058B90C0B48C6BE5DF846C5B2EF) >> 128;
            if (x & 0x4000000000 > 0)
                result = (result * 0x10000002C5C8601CC6B9E94213C72737A) >> 128;
            if (x & 0x2000000000 > 0)
                result = (result * 0x1000000162E42FFF037DF38AA2B219F06) >> 128;
            if (x & 0x1000000000 > 0)
                result = (result * 0x10000000B17217FBA9C739AA5819F44F9) >> 128;
            if (x & 0x800000000 > 0)
                result = (result * 0x1000000058B90BFCDEE5ACD3C1CEDC823) >> 128;
            if (x & 0x400000000 > 0)
                result = (result * 0x100000002C5C85FE31F35A6A30DA1BE50) >> 128;
            if (x & 0x200000000 > 0)
                result = (result * 0x10000000162E42FF0999CE3541B9FFFCF) >> 128;
            if (x & 0x100000000 > 0)
                result = (result * 0x100000000B17217F80F4EF5AADDA45554) >> 128;
            if (x & 0x80000000 > 0)
                result = (result * 0x10000000058B90BFBF8479BD5A81B51AD) >> 128;
            if (x & 0x40000000 > 0)
                result = (result * 0x1000000002C5C85FDF84BD62AE30A74CC) >> 128;
            if (x & 0x20000000 > 0)
                result = (result * 0x100000000162E42FEFB2FED257559BDAA) >> 128;
            if (x & 0x10000000 > 0)
                result = (result * 0x1000000000B17217F7D5A7716BBA4A9AE) >> 128;
            if (x & 0x8000000 > 0)
                result = (result * 0x100000000058B90BFBE9DDBAC5E109CCE) >> 128;
            if (x & 0x4000000 > 0)
                result = (result * 0x10000000002C5C85FDF4B15DE6F17EB0D) >> 128;
            if (x & 0x2000000 > 0)
                result = (result * 0x1000000000162E42FEFA494F1478FDE05) >> 128;
            if (x & 0x1000000 > 0)
                result = (result * 0x10000000000B17217F7D20CF927C8E94C) >> 128;
            if (x & 0x800000 > 0)
                result = (result * 0x1000000000058B90BFBE8F71CB4E4B33D) >> 128;
            if (x & 0x400000 > 0)
                result = (result * 0x100000000002C5C85FDF477B662B26945) >> 128;
            if (x & 0x200000 > 0)
                result = (result * 0x10000000000162E42FEFA3AE53369388C) >> 128;
            if (x & 0x100000 > 0)
                result = (result * 0x100000000000B17217F7D1D351A389D40) >> 128;
            if (x & 0x80000 > 0)
                result = (result * 0x10000000000058B90BFBE8E8B2D3D4EDE) >> 128;
            if (x & 0x40000 > 0)
                result = (result * 0x1000000000002C5C85FDF4741BEA6E77E) >> 128;
            if (x & 0x20000 > 0)
                result = (result * 0x100000000000162E42FEFA39FE95583C2) >> 128;
            if (x & 0x10000 > 0)
                result = (result * 0x1000000000000B17217F7D1CFB72B45E1) >> 128;
            if (x & 0x8000 > 0)
                result = (result * 0x100000000000058B90BFBE8E7CC35C3F0) >> 128;
            if (x & 0x4000 > 0)
                result = (result * 0x10000000000002C5C85FDF473E242EA38) >> 128;
            if (x & 0x2000 > 0)
                result = (result * 0x1000000000000162E42FEFA39F02B772C) >> 128;
            if (x & 0x1000 > 0)
                result = (result * 0x10000000000000B17217F7D1CF7D83C1A) >> 128;
            if (x & 0x800 > 0)
                result = (result * 0x1000000000000058B90BFBE8E7BDCBE2E) >> 128;
            if (x & 0x400 > 0)
                result = (result * 0x100000000000002C5C85FDF473DEA871F) >> 128;
            if (x & 0x200 > 0)
                result = (result * 0x10000000000000162E42FEFA39EF44D91) >> 128;
            if (x & 0x100 > 0)
                result = (result * 0x100000000000000B17217F7D1CF79E949) >> 128;
            if (x & 0x80 > 0)
                result = (result * 0x10000000000000058B90BFBE8E7BCE544) >> 128;
            if (x & 0x40 > 0)
                result = (result * 0x1000000000000002C5C85FDF473DE6ECA) >> 128;
            if (x & 0x20 > 0)
                result = (result * 0x100000000000000162E42FEFA39EF366F) >> 128;
            if (x & 0x10 > 0)
                result = (result * 0x1000000000000000B17217F7D1CF79AFA) >> 128;
            if (x & 0x8 > 0)
                result = (result * 0x100000000000000058B90BFBE8E7BCD6D) >> 128;
            if (x & 0x4 > 0)
                result = (result * 0x10000000000000002C5C85FDF473DE6B2) >> 128;
            if (x & 0x2 > 0)
                result = (result * 0x1000000000000000162E42FEFA39EF358) >> 128;
            if (x & 0x1 > 0)
                result = (result * 0x10000000000000000B17217F7D1CF79AB) >> 128;

            result >>= uint256(int256(63 - (x >> 64)));
            require(result <= uint256(int256(MAX_64x64)));

            return int128(int256(result));
        }
    }

    /**
     * Calculate natural exponent of x.  Revert on overflow.
     *
     * @param x signed 64.64-bit fixed point number
     * @return signed 64.64-bit fixed point number
     */
    function exp(int128 x) internal pure returns (int128) {
        unchecked {
            require(x < 0x400000000000000000); // Overflow

            if (x < -0x400000000000000000) return 0; // Underflow

            return
                exp_2(
                    int128(
                        (int256(x) * 0x171547652B82FE1777D0FFDA0D23A7D12) >> 128
                    )
                );
        }
    }

    /**
     * Calculate x / y rounding towards zero, where x and y are unsigned 256-bit
     * integer numbers.  Revert on overflow or when y is zero.
     *
     * @param x unsigned 256-bit integer number
     * @param y unsigned 256-bit integer number
     * @return unsigned 64.64-bit fixed point number
     */
    function divuu(uint256 x, uint256 y) private pure returns (uint128) {
        unchecked {
            require(y != 0);

            uint256 result;

            if (x <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF)
                result = (x << 64) / y;
            else {
                uint256 msb = 192;
                uint256 xc = x >> 192;
                if (xc >= 0x100000000) {
                    xc >>= 32;
                    msb += 32;
                }
                if (xc >= 0x10000) {
                    xc >>= 16;
                    msb += 16;
                }
                if (xc >= 0x100) {
                    xc >>= 8;
                    msb += 8;
                }
                if (xc >= 0x10) {
                    xc >>= 4;
                    msb += 4;
                }
                if (xc >= 0x4) {
                    xc >>= 2;
                    msb += 2;
                }
                if (xc >= 0x2) msb += 1; // No need to shift xc anymore

                result = (x << (255 - msb)) / (((y - 1) >> (msb - 191)) + 1);
                require(result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);

                uint256 hi = result * (y >> 128);
                uint256 lo = result * (y & 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);

                uint256 xh = x >> 192;
                uint256 xl = x << 64;

                if (xl < lo) xh -= 1;
                xl -= lo; // We rely on overflow behavior here
                lo = hi << 128;
                if (xl < lo) xh -= 1;
                xl -= lo; // We rely on overflow behavior here

                assert(xh == hi >> 128);

                result += xl / y;
            }

            require(result <= 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF);
            return uint128(result);
        }
    }

    /**
     * Calculate sqrt (x) rounding down, where x is unsigned 256-bit integer
     * number.
     *
     * @param x unsigned 256-bit integer number
     * @return unsigned 128-bit integer number
     */
    function sqrtu(uint256 x) private pure returns (uint128) {
        unchecked {
            if (x == 0) return 0;
            else {
                uint256 xx = x;
                uint256 r = 1;
                if (xx >= 0x100000000000000000000000000000000) {
                    xx >>= 128;
                    r <<= 64;
                }
                if (xx >= 0x10000000000000000) {
                    xx >>= 64;
                    r <<= 32;
                }
                if (xx >= 0x100000000) {
                    xx >>= 32;
                    r <<= 16;
                }
                if (xx >= 0x10000) {
                    xx >>= 16;
                    r <<= 8;
                }
                if (xx >= 0x100) {
                    xx >>= 8;
                    r <<= 4;
                }
                if (xx >= 0x10) {
                    xx >>= 4;
                    r <<= 2;
                }
                if (xx >= 0x8) {
                    r <<= 1;
                }
                r = (r + x / r) >> 1;
                r = (r + x / r) >> 1;
                r = (r + x / r) >> 1;
                r = (r + x / r) >> 1;
                r = (r + x / r) >> 1;
                r = (r + x / r) >> 1;
                r = (r + x / r) >> 1; // Seven iterations should be enough
                uint256 r1 = x / r;
                return uint128(r < r1 ? r : r1);
            }
        }
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/security/ReentrancyGuard.sol

// OpenZeppelin Contracts v4.4.1 (security/ReentrancyGuard.sol)

pragma solidity ^0.8.0;

/**
 * @dev Contract module that helps prevent reentrant calls to a function.
 *
 * Inheriting from `ReentrancyGuard` will make the {nonReentrant} modifier
 * available, which can be applied to functions to make sure there are no nested
 * (reentrant) calls to them.
 *
 * Note that because there is a single `nonReentrant` guard, functions marked as
 * `nonReentrant` may not call one another. This can be worked around by making
 * those functions `private`, and then adding `external` `nonReentrant` entry
 * points to them.
 *
 * TIP: If you would like to learn more about reentrancy and alternative ways
 * to protect against it, check out our blog post
 * https://blog.openzeppelin.com/reentrancy-after-istanbul/[Reentrancy After Istanbul].
 */
abstract contract ReentrancyGuard {
    // Booleans are more expensive than uint256 or any type that takes up a full
    // word because each write operation emits an extra SLOAD to first read the
    // slot's contents, replace the bits taken up by the boolean, and then write
    // back. This is the compiler's defense against contract upgrades and
    // pointer aliasing, and it cannot be disabled.

    // The values being non-zero value makes deployment a bit more expensive,
    // but in exchange the refund on every call to nonReentrant will be lower in
    // amount. Since refunds are capped to a percentage of the total
    // transaction's gas, it is best to keep them low in cases like this one, to
    // increase the likelihood of the full refund coming into effect.
    uint256 private constant _NOT_ENTERED = 1;
    uint256 private constant _ENTERED = 2;

    uint256 private _status;

    constructor() {
        _status = _NOT_ENTERED;
    }

    /**
     * @dev Prevents a contract from calling itself, directly or indirectly.
     * Calling a `nonReentrant` function from another `nonReentrant`
     * function is not supported. It is possible to prevent this from happening
     * by making the `nonReentrant` function external, and making it call a
     * `private` function that does the actual work.
     */
    modifier nonReentrant() {
        _nonReentrantBefore();
        _;
        _nonReentrantAfter();
    }

    function _nonReentrantBefore() private {
        // On the first call to nonReentrant, _notEntered will be true
        require(_status != _ENTERED, "ReentrancyGuard: reentrant call");

        // Any calls to nonReentrant after this point will fail
        _status = _ENTERED;
    }

    function _nonReentrantAfter() private {
        // By storing the original value once again, a refund is triggered (see
        // https://eips.ethereum.org/EIPS/eip-2200)
        _status = _NOT_ENTERED;
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/utils/Context.sol

// OpenZeppelin Contracts v4.4.1 (utils/Context.sol)

pragma solidity ^0.8.0;

/**
 * @dev Provides information about the current execution context, including the
 * sender of the transaction and its data. While these are generally available
 * via msg.sender and msg.data, they should not be accessed in such a direct
 * manner, since when dealing with meta-transactions the account sending and
 * paying for execution may not be the actual sender (as far as an application
 * is concerned).
 *
 * This contract is only required for intermediate, library-like contracts.
 */
abstract contract Context {
    function _msgSender() internal view virtual returns (address) {
        return msg.sender;
    }

    function _msgData() internal view virtual returns (bytes calldata) {
        return msg.data;
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/access/Ownable.sol

// OpenZeppelin Contracts (last updated v4.7.0) (access/Ownable.sol)

pragma solidity ^0.8.0;

/**
 * @dev Contract module which provides a basic access control mechanism, where
 * there is an account (an owner) that can be granted exclusive access to
 * specific functions.
 *
 * By default, the owner account will be the one that deploys the contract. This
 * can later be changed with {transferOwnership}.
 *
 * This module is used through inheritance. It will make available the modifier
 * `onlyOwner`, which can be applied to your functions to restrict their use to
 * the owner.
 */
abstract contract Ownable is Context {
    address private _owner;

    event OwnershipTransferred(
        address indexed previousOwner,
        address indexed newOwner
    );

    /**
     * @dev Initializes the contract setting the deployer as the initial owner.
     */
    constructor() {
        _transferOwnership(_msgSender());
    }

    /**
     * @dev Throws if called by any account other than the owner.
     */
    modifier onlyOwner() {
        _checkOwner();
        _;
    }

    /**
     * @dev Returns the address of the current owner.
     */
    function owner() public view virtual returns (address) {
        return _owner;
    }

    /**
     * @dev Throws if the sender is not the owner.
     */
    function _checkOwner() internal view virtual {
        require(owner() == _msgSender(), "Ownable: caller is not the owner");
    }

    /**
     * @dev Leaves the contract without owner. It will not be possible to call
     * `onlyOwner` functions anymore. Can only be called by the current owner.
     *
     * NOTE: Renouncing ownership will leave the contract without an owner,
     * thereby removing any functionality that is only available to the owner.
     */
    function renounceOwnership() public virtual onlyOwner {
        _transferOwnership(address(0));
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Can only be called by the current owner.
     */
    function transferOwnership(address newOwner) public virtual onlyOwner {
        require(
            newOwner != address(0),
            "Ownable: new owner is the zero address"
        );
        _transferOwnership(newOwner);
    }

    /**
     * @dev Transfers ownership of the contract to a new account (`newOwner`).
     * Internal function without access restriction.
     */
    function _transferOwnership(address newOwner) internal virtual {
        address oldOwner = _owner;
        _owner = newOwner;
        emit OwnershipTransferred(oldOwner, newOwner);
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/utils/math/SafeMath.sol

// OpenZeppelin Contracts (last updated v4.6.0) (utils/math/SafeMath.sol)

pragma solidity ^0.8.0;

// CAUTION
// This version of SafeMath should only be used with Solidity 0.8 or later,
// because it relies on the compiler's built in overflow checks.

/**
 * @dev Wrappers over Solidity's arithmetic operations.
 *
 * NOTE: `SafeMath` is generally not needed starting with Solidity 0.8, since the compiler
 * now has built in overflow checking.
 */
library SafeMath {
    /**
     * @dev Returns the addition of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function tryAdd(uint256 a, uint256 b)
        internal
        pure
        returns (bool, uint256)
    {
        unchecked {
            uint256 c = a + b;
            if (c < a) return (false, 0);
            return (true, c);
        }
    }

    /**
     * @dev Returns the subtraction of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function trySub(uint256 a, uint256 b)
        internal
        pure
        returns (bool, uint256)
    {
        unchecked {
            if (b > a) return (false, 0);
            return (true, a - b);
        }
    }

    /**
     * @dev Returns the multiplication of two unsigned integers, with an overflow flag.
     *
     * _Available since v3.4._
     */
    function tryMul(uint256 a, uint256 b)
        internal
        pure
        returns (bool, uint256)
    {
        unchecked {
            // Gas optimization: this is cheaper than requiring 'a' not being zero, but the
            // benefit is lost if 'b' is also tested.
            // See: https://github.com/OpenZeppelin/openzeppelin-contracts/pull/522
            if (a == 0) return (true, 0);
            uint256 c = a * b;
            if (c / a != b) return (false, 0);
            return (true, c);
        }
    }

    /**
     * @dev Returns the division of two unsigned integers, with a division by zero flag.
     *
     * _Available since v3.4._
     */
    function tryDiv(uint256 a, uint256 b)
        internal
        pure
        returns (bool, uint256)
    {
        unchecked {
            if (b == 0) return (false, 0);
            return (true, a / b);
        }
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers, with a division by zero flag.
     *
     * _Available since v3.4._
     */
    function tryMod(uint256 a, uint256 b)
        internal
        pure
        returns (bool, uint256)
    {
        unchecked {
            if (b == 0) return (false, 0);
            return (true, a % b);
        }
    }

    /**
     * @dev Returns the addition of two unsigned integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `+` operator.
     *
     * Requirements:
     *
     * - Addition cannot overflow.
     */
    function add(uint256 a, uint256 b) internal pure returns (uint256) {
        return a + b;
    }

    /**
     * @dev Returns the subtraction of two unsigned integers, reverting on
     * overflow (when the result is negative).
     *
     * Counterpart to Solidity's `-` operator.
     *
     * Requirements:
     *
     * - Subtraction cannot overflow.
     */
    function sub(uint256 a, uint256 b) internal pure returns (uint256) {
        return a - b;
    }

    /**
     * @dev Returns the multiplication of two unsigned integers, reverting on
     * overflow.
     *
     * Counterpart to Solidity's `*` operator.
     *
     * Requirements:
     *
     * - Multiplication cannot overflow.
     */
    function mul(uint256 a, uint256 b) internal pure returns (uint256) {
        return a * b;
    }

    /**
     * @dev Returns the integer division of two unsigned integers, reverting on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `/` operator.
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(uint256 a, uint256 b) internal pure returns (uint256) {
        return a / b;
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),
     * reverting when dividing by zero.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function mod(uint256 a, uint256 b) internal pure returns (uint256) {
        return a % b;
    }

    /**
     * @dev Returns the subtraction of two unsigned integers, reverting with custom message on
     * overflow (when the result is negative).
     *
     * CAUTION: This function is deprecated because it requires allocating memory for the error
     * message unnecessarily. For custom revert reasons use {trySub}.
     *
     * Counterpart to Solidity's `-` operator.
     *
     * Requirements:
     *
     * - Subtraction cannot overflow.
     */
    function sub(
        uint256 a,
        uint256 b,
        string memory errorMessage
    ) internal pure returns (uint256) {
        unchecked {
            require(b <= a, errorMessage);
            return a - b;
        }
    }

    /**
     * @dev Returns the integer division of two unsigned integers, reverting with custom message on
     * division by zero. The result is rounded towards zero.
     *
     * Counterpart to Solidity's `/` operator. Note: this function uses a
     * `revert` opcode (which leaves remaining gas untouched) while Solidity
     * uses an invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function div(
        uint256 a,
        uint256 b,
        string memory errorMessage
    ) internal pure returns (uint256) {
        unchecked {
            require(b > 0, errorMessage);
            return a / b;
        }
    }

    /**
     * @dev Returns the remainder of dividing two unsigned integers. (unsigned integer modulo),
     * reverting with custom message when dividing by zero.
     *
     * CAUTION: This function is deprecated because it requires allocating memory for the error
     * message unnecessarily. For custom revert reasons use {tryMod}.
     *
     * Counterpart to Solidity's `%` operator. This function uses a `revert`
     * opcode (which leaves remaining gas untouched) while Solidity uses an
     * invalid opcode to revert (consuming all remaining gas).
     *
     * Requirements:
     *
     * - The divisor cannot be zero.
     */
    function mod(
        uint256 a,
        uint256 b,
        string memory errorMessage
    ) internal pure returns (uint256) {
        unchecked {
            require(b > 0, errorMessage);
            return a % b;
        }
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/utils/structs/EnumerableSet.sol

// OpenZeppelin Contracts (last updated v4.7.0) (utils/structs/EnumerableSet.sol)
// This file was procedurally generated from scripts/generate/templates/EnumerableSet.js.

pragma solidity ^0.8.0;

/**
 * @dev Library for managing
 * https://en.wikipedia.org/wiki/Set_(abstract_data_type)[sets] of primitive
 * types.
 *
 * Sets have the following properties:
 *
 * - Elements are added, removed, and checked for existence in constant time
 * (O(1)).
 * - Elements are enumerated in O(n). No guarantees are made on the ordering.
 *
 * ```
 * contract Example {
 *     // Add the library methods
 *     using EnumerableSet for EnumerableSet.AddressSet;
 *
 *     // Declare a set state variable
 *     EnumerableSet.AddressSet private mySet;
 * }
 * ```
 *
 * As of v3.3.0, sets of type `bytes32` (`Bytes32Set`), `address` (`AddressSet`)
 * and `uint256` (`UintSet`) are supported.
 *
 * [WARNING]
 * ====
 * Trying to delete such a structure from storage will likely result in data corruption, rendering the structure
 * unusable.
 * See https://github.com/ethereum/solidity/pull/11843[ethereum/solidity#11843] for more info.
 *
 * In order to clean an EnumerableSet, you can either remove all elements one by one or create a fresh instance using an
 * array of EnumerableSet.
 * ====
 */
library EnumerableSet {
    // To implement this library for multiple types with as little code
    // repetition as possible, we write it in terms of a generic Set type with
    // bytes32 values.
    // The Set implementation uses private functions, and user-facing
    // implementations (such as AddressSet) are just wrappers around the
    // underlying Set.
    // This means that we can only create new EnumerableSets for types that fit
    // in bytes32.

    struct Set {
        // Storage of set values
        bytes32[] _values;
        // Position of the value in the `values` array, plus 1 because index 0
        // means a value is not in the set.
        mapping(bytes32 => uint256) _indexes;
    }

    /**
     * @dev Add a value to a set. O(1).
     *
     * Returns true if the value was added to the set, that is if it was not
     * already present.
     */
    function _add(Set storage set, bytes32 value) private returns (bool) {
        if (!_contains(set, value)) {
            set._values.push(value);
            // The value is stored at length-1, but we add 1 to all indexes
            // and use 0 as a sentinel value
            set._indexes[value] = set._values.length;
            return true;
        } else {
            return false;
        }
    }

    /**
     * @dev Removes a value from a set. O(1).
     *
     * Returns true if the value was removed from the set, that is if it was
     * present.
     */
    function _remove(Set storage set, bytes32 value) private returns (bool) {
        // We read and store the value's index to prevent multiple reads from the same storage slot
        uint256 valueIndex = set._indexes[value];

        if (valueIndex != 0) {
            // Equivalent to contains(set, value)
            // To delete an element from the _values array in O(1), we swap the element to delete with the last one in
            // the array, and then remove the last element (sometimes called as 'swap and pop').
            // This modifies the order of the array, as noted in {at}.

            uint256 toDeleteIndex = valueIndex - 1;
            uint256 lastIndex = set._values.length - 1;

            if (lastIndex != toDeleteIndex) {
                bytes32 lastValue = set._values[lastIndex];

                // Move the last value to the index where the value to delete is
                set._values[toDeleteIndex] = lastValue;
                // Update the index for the moved value
                set._indexes[lastValue] = valueIndex; // Replace lastValue's index to valueIndex
            }

            // Delete the slot where the moved value was stored
            set._values.pop();

            // Delete the index for the deleted slot
            delete set._indexes[value];

            return true;
        } else {
            return false;
        }
    }

    /**
     * @dev Returns true if the value is in the set. O(1).
     */
    function _contains(Set storage set, bytes32 value)
        private
        view
        returns (bool)
    {
        return set._indexes[value] != 0;
    }

    /**
     * @dev Returns the number of values on the set. O(1).
     */
    function _length(Set storage set) private view returns (uint256) {
        return set._values.length;
    }

    /**
     * @dev Returns the value stored at position `index` in the set. O(1).
     *
     * Note that there are no guarantees on the ordering of values inside the
     * array, and it may change when more values are added or removed.
     *
     * Requirements:
     *
     * - `index` must be strictly less than {length}.
     */
    function _at(Set storage set, uint256 index)
        private
        view
        returns (bytes32)
    {
        return set._values[index];
    }

    /**
     * @dev Return the entire set in an array
     *
     * WARNING: This operation will copy the entire storage to memory, which can be quite expensive. This is designed
     * to mostly be used by view accessors that are queried without any gas fees. Developers should keep in mind that
     * this function has an unbounded cost, and using it as part of a state-changing function may render the function
     * uncallable if the set grows to a point where copying to memory consumes too much gas to fit in a block.
     */
    function _values(Set storage set) private view returns (bytes32[] memory) {
        return set._values;
    }

    // Bytes32Set

    struct Bytes32Set {
        Set _inner;
    }

    /**
     * @dev Add a value to a set. O(1).
     *
     * Returns true if the value was added to the set, that is if it was not
     * already present.
     */
    function add(Bytes32Set storage set, bytes32 value)
        internal
        returns (bool)
    {
        return _add(set._inner, value);
    }

    /**
     * @dev Removes a value from a set. O(1).
     *
     * Returns true if the value was removed from the set, that is if it was
     * present.
     */
    function remove(Bytes32Set storage set, bytes32 value)
        internal
        returns (bool)
    {
        return _remove(set._inner, value);
    }

    /**
     * @dev Returns true if the value is in the set. O(1).
     */
    function contains(Bytes32Set storage set, bytes32 value)
        internal
        view
        returns (bool)
    {
        return _contains(set._inner, value);
    }

    /**
     * @dev Returns the number of values in the set. O(1).
     */
    function length(Bytes32Set storage set) internal view returns (uint256) {
        return _length(set._inner);
    }

    /**
     * @dev Returns the value stored at position `index` in the set. O(1).
     *
     * Note that there are no guarantees on the ordering of values inside the
     * array, and it may change when more values are added or removed.
     *
     * Requirements:
     *
     * - `index` must be strictly less than {length}.
     */
    function at(Bytes32Set storage set, uint256 index)
        internal
        view
        returns (bytes32)
    {
        return _at(set._inner, index);
    }

    /**
     * @dev Return the entire set in an array
     *
     * WARNING: This operation will copy the entire storage to memory, which can be quite expensive. This is designed
     * to mostly be used by view accessors that are queried without any gas fees. Developers should keep in mind that
     * this function has an unbounded cost, and using it as part of a state-changing function may render the function
     * uncallable if the set grows to a point where copying to memory consumes too much gas to fit in a block.
     */
    function values(Bytes32Set storage set)
        internal
        view
        returns (bytes32[] memory)
    {
        bytes32[] memory store = _values(set._inner);
        bytes32[] memory result;

        /// @solidity memory-safe-assembly
        assembly {
            result := store
        }

        return result;
    }

    // AddressSet

    struct AddressSet {
        Set _inner;
    }

    /**
     * @dev Add a value to a set. O(1).
     *
     * Returns true if the value was added to the set, that is if it was not
     * already present.
     */
    function add(AddressSet storage set, address value)
        internal
        returns (bool)
    {
        return _add(set._inner, bytes32(uint256(uint160(value))));
    }

    /**
     * @dev Removes a value from a set. O(1).
     *
     * Returns true if the value was removed from the set, that is if it was
     * present.
     */
    function remove(AddressSet storage set, address value)
        internal
        returns (bool)
    {
        return _remove(set._inner, bytes32(uint256(uint160(value))));
    }

    /**
     * @dev Returns true if the value is in the set. O(1).
     */
    function contains(AddressSet storage set, address value)
        internal
        view
        returns (bool)
    {
        return _contains(set._inner, bytes32(uint256(uint160(value))));
    }

    /**
     * @dev Returns the number of values in the set. O(1).
     */
    function length(AddressSet storage set) internal view returns (uint256) {
        return _length(set._inner);
    }

    /**
     * @dev Returns the value stored at position `index` in the set. O(1).
     *
     * Note that there are no guarantees on the ordering of values inside the
     * array, and it may change when more values are added or removed.
     *
     * Requirements:
     *
     * - `index` must be strictly less than {length}.
     */
    function at(AddressSet storage set, uint256 index)
        internal
        view
        returns (address)
    {
        return address(uint160(uint256(_at(set._inner, index))));
    }

    /**
     * @dev Return the entire set in an array
     *
     * WARNING: This operation will copy the entire storage to memory, which can be quite expensive. This is designed
     * to mostly be used by view accessors that are queried without any gas fees. Developers should keep in mind that
     * this function has an unbounded cost, and using it as part of a state-changing function may render the function
     * uncallable if the set grows to a point where copying to memory consumes too much gas to fit in a block.
     */
    function values(AddressSet storage set)
        internal
        view
        returns (address[] memory)
    {
        bytes32[] memory store = _values(set._inner);
        address[] memory result;

        /// @solidity memory-safe-assembly
        assembly {
            result := store
        }

        return result;
    }

    // UintSet

    struct UintSet {
        Set _inner;
    }

    /**
     * @dev Add a value to a set. O(1).
     *
     * Returns true if the value was added to the set, that is if it was not
     * already present.
     */
    function add(UintSet storage set, uint256 value) internal returns (bool) {
        return _add(set._inner, bytes32(value));
    }

    /**
     * @dev Removes a value from a set. O(1).
     *
     * Returns true if the value was removed from the set, that is if it was
     * present.
     */
    function remove(UintSet storage set, uint256 value)
        internal
        returns (bool)
    {
        return _remove(set._inner, bytes32(value));
    }

    /**
     * @dev Returns true if the value is in the set. O(1).
     */
    function contains(UintSet storage set, uint256 value)
        internal
        view
        returns (bool)
    {
        return _contains(set._inner, bytes32(value));
    }

    /**
     * @dev Returns the number of values in the set. O(1).
     */
    function length(UintSet storage set) internal view returns (uint256) {
        return _length(set._inner);
    }

    /**
     * @dev Returns the value stored at position `index` in the set. O(1).
     *
     * Note that there are no guarantees on the ordering of values inside the
     * array, and it may change when more values are added or removed.
     *
     * Requirements:
     *
     * - `index` must be strictly less than {length}.
     */
    function at(UintSet storage set, uint256 index)
        internal
        view
        returns (uint256)
    {
        return uint256(_at(set._inner, index));
    }

    /**
     * @dev Return the entire set in an array
     *
     * WARNING: This operation will copy the entire storage to memory, which can be quite expensive. This is designed
     * to mostly be used by view accessors that are queried without any gas fees. Developers should keep in mind that
     * this function has an unbounded cost, and using it as part of a state-changing function may render the function
     * uncallable if the set grows to a point where copying to memory consumes too much gas to fit in a block.
     */
    function values(UintSet storage set)
        internal
        view
        returns (uint256[] memory)
    {
        bytes32[] memory store = _values(set._inner);
        uint256[] memory result;

        /// @solidity memory-safe-assembly
        assembly {
            result := store
        }

        return result;
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/utils/Address.sol

// OpenZeppelin Contracts (last updated v4.7.0) (utils/Address.sol)

pragma solidity ^0.8.1;

/**
 * @dev Collection of functions related to the address type
 */
library Address {
    /**
     * @dev Returns true if `account` is a contract.
     *
     * [IMPORTANT]
     * ====
     * It is unsafe to assume that an address for which this function returns
     * false is an externally-owned account (EOA) and not a contract.
     *
     * Among others, `isContract` will return false for the following
     * types of addresses:
     *
     *  - an externally-owned account
     *  - a contract in construction
     *  - an address where a contract will be created
     *  - an address where a contract lived, but was destroyed
     * ====
     *
     * [IMPORTANT]
     * ====
     * You shouldn't rely on `isContract` to protect against flash loan attacks!
     *
     * Preventing calls from contracts is highly discouraged. It breaks composability, breaks support for smart wallets
     * like Gnosis Safe, and does not provide security since it can be circumvented by calling from a contract
     * constructor.
     * ====
     */
    function isContract(address account) internal view returns (bool) {
        // This method relies on extcodesize/address.code.length, which returns 0
        // for contracts in construction, since the code is only stored at the end
        // of the constructor execution.

        return account.code.length > 0;
    }

    /**
     * @dev Replacement for Solidity's `transfer`: sends `amount` wei to
     * `recipient`, forwarding all available gas and reverting on errors.
     *
     * https://eips.ethereum.org/EIPS/eip-1884[EIP1884] increases the gas cost
     * of certain opcodes, possibly making contracts go over the 2300 gas limit
     * imposed by `transfer`, making them unable to receive funds via
     * `transfer`. {sendValue} removes this limitation.
     *
     * https://diligence.consensys.net/posts/2019/09/stop-using-soliditys-transfer-now/[Learn more].
     *
     * IMPORTANT: because control is transferred to `recipient`, care must be
     * taken to not create reentrancy vulnerabilities. Consider using
     * {ReentrancyGuard} or the
     * https://solidity.readthedocs.io/en/v0.5.11/security-considerations.html#use-the-checks-effects-interactions-pattern[checks-effects-interactions pattern].
     */
    function sendValue(address payable recipient, uint256 amount) internal {
        require(
            address(this).balance >= amount,
            "Address: insufficient balance"
        );

        (bool success, ) = recipient.call{value: amount}("");
        require(
            success,
            "Address: unable to send value, recipient may have reverted"
        );
    }

    /**
     * @dev Performs a Solidity function call using a low level `call`. A
     * plain `call` is an unsafe replacement for a function call: use this
     * function instead.
     *
     * If `target` reverts with a revert reason, it is bubbled up by this
     * function (like regular Solidity function calls).
     *
     * Returns the raw returned data. To convert to the expected return value,
     * use https://solidity.readthedocs.io/en/latest/units-and-global-variables.html?highlight=abi.decode#abi-encoding-and-decoding-functions[`abi.decode`].
     *
     * Requirements:
     *
     * - `target` must be a contract.
     * - calling `target` with `data` must not revert.
     *
     * _Available since v3.1._
     */
    function functionCall(address target, bytes memory data)
        internal
        returns (bytes memory)
    {
        return
            functionCallWithValue(
                target,
                data,
                0,
                "Address: low-level call failed"
            );
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`], but with
     * `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal returns (bytes memory) {
        return functionCallWithValue(target, data, 0, errorMessage);
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but also transferring `value` wei to `target`.
     *
     * Requirements:
     *
     * - the calling contract must have an ETH balance of at least `value`.
     * - the called Solidity function must be `payable`.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(
        address target,
        bytes memory data,
        uint256 value
    ) internal returns (bytes memory) {
        return
            functionCallWithValue(
                target,
                data,
                value,
                "Address: low-level call with value failed"
            );
    }

    /**
     * @dev Same as {xref-Address-functionCallWithValue-address-bytes-uint256-}[`functionCallWithValue`], but
     * with `errorMessage` as a fallback revert reason when `target` reverts.
     *
     * _Available since v3.1._
     */
    function functionCallWithValue(
        address target,
        bytes memory data,
        uint256 value,
        string memory errorMessage
    ) internal returns (bytes memory) {
        require(
            address(this).balance >= value,
            "Address: insufficient balance for call"
        );
        (bool success, bytes memory returndata) = target.call{value: value}(
            data
        );
        return
            verifyCallResultFromTarget(
                target,
                success,
                returndata,
                errorMessage
            );
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(address target, bytes memory data)
        internal
        view
        returns (bytes memory)
    {
        return
            functionStaticCall(
                target,
                data,
                "Address: low-level static call failed"
            );
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a static call.
     *
     * _Available since v3.3._
     */
    function functionStaticCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal view returns (bytes memory) {
        (bool success, bytes memory returndata) = target.staticcall(data);
        return
            verifyCallResultFromTarget(
                target,
                success,
                returndata,
                errorMessage
            );
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(address target, bytes memory data)
        internal
        returns (bytes memory)
    {
        return
            functionDelegateCall(
                target,
                data,
                "Address: low-level delegate call failed"
            );
    }

    /**
     * @dev Same as {xref-Address-functionCall-address-bytes-string-}[`functionCall`],
     * but performing a delegate call.
     *
     * _Available since v3.4._
     */
    function functionDelegateCall(
        address target,
        bytes memory data,
        string memory errorMessage
    ) internal returns (bytes memory) {
        (bool success, bytes memory returndata) = target.delegatecall(data);
        return
            verifyCallResultFromTarget(
                target,
                success,
                returndata,
                errorMessage
            );
    }

    /**
     * @dev Tool to verify that a low level call to smart-contract was successful, and revert (either by bubbling
     * the revert reason or using the provided one) in case of unsuccessful call or if target was not a contract.
     *
     * _Available since v4.8._
     */
    function verifyCallResultFromTarget(
        address target,
        bool success,
        bytes memory returndata,
        string memory errorMessage
    ) internal view returns (bytes memory) {
        if (success) {
            if (returndata.length == 0) {
                // only check isContract if the call was successful and the return data is empty
                // otherwise we already know that it was a contract
                require(isContract(target), "Address: call to non-contract");
            }
            return returndata;
        } else {
            _revert(returndata, errorMessage);
        }
    }

    /**
     * @dev Tool to verify that a low level call was successful, and revert if it wasn't, either by bubbling the
     * revert reason or using the provided one.
     *
     * _Available since v4.3._
     */
    function verifyCallResult(
        bool success,
        bytes memory returndata,
        string memory errorMessage
    ) internal pure returns (bytes memory) {
        if (success) {
            return returndata;
        } else {
            _revert(returndata, errorMessage);
        }
    }

    function _revert(bytes memory returndata, string memory errorMessage)
        private
        pure
    {
        // Look for revert reason and bubble it up if present
        if (returndata.length > 0) {
            // The easiest way to bubble the revert reason is using memory via assembly
            /// @solidity memory-safe-assembly
            assembly {
                let returndata_size := mload(returndata)
                revert(add(32, returndata), returndata_size)
            }
        } else {
            revert(errorMessage);
        }
    }
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/token/ERC20/extensions/draft-IERC20Permit.sol

// OpenZeppelin Contracts v4.4.1 (token/ERC20/extensions/draft-IERC20Permit.sol)

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 Permit extension allowing approvals to be made via signatures, as defined in
 * https://eips.ethereum.org/EIPS/eip-2612[EIP-2612].
 *
 * Adds the {permit} method, which can be used to change an account's ERC20 allowance (see {IERC20-allowance}) by
 * presenting a message signed by the account. By not relying on {IERC20-approve}, the token holder account doesn't
 * need to send a transaction, and thus is not required to hold Ether at all.
 */
interface IERC20Permit {
    /**
     * @dev Sets `value` as the allowance of `spender` over ``owner``'s tokens,
     * given ``owner``'s signed approval.
     *
     * IMPORTANT: The same issues {IERC20-approve} has related to transaction
     * ordering also apply here.
     *
     * Emits an {Approval} event.
     *
     * Requirements:
     *
     * - `spender` cannot be the zero address.
     * - `deadline` must be a timestamp in the future.
     * - `v`, `r` and `s` must be a valid `secp256k1` signature from `owner`
     * over the EIP712-formatted function arguments.
     * - the signature must use ``owner``'s current nonce (see {nonces}).
     *
     * For more information on the signature format, see the
     * https://eips.ethereum.org/EIPS/eip-2612#specification[relevant EIP
     * section].
     */
    function permit(
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) external;

    /**
     * @dev Returns the current nonce for `owner`. This value must be
     * included whenever a signature is generated for {permit}.
     *
     * Every successful call to {permit} increases ``owner``'s nonce by one. This
     * prevents a signature from being used multiple times.
     */
    function nonces(address owner) external view returns (uint256);

    /**
     * @dev Returns the domain separator used in the encoding of the signature for {permit}, as defined by {EIP712}.
     */
    // solhint-disable-next-line func-name-mixedcase
    function DOMAIN_SEPARATOR() external view returns (bytes32);
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/token/ERC20/IERC20.sol

// OpenZeppelin Contracts (last updated v4.6.0) (token/ERC20/IERC20.sol)

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 standard as defined in the EIP.
 */
interface IERC20 {
    /**
     * @dev Emitted when `value` tokens are moved from one account (`from`) to
     * another (`to`).
     *
     * Note that `value` may be zero.
     */
    event Transfer(address indexed from, address indexed to, uint256 value);

    /**
     * @dev Emitted when the allowance of a `spender` for an `owner` is set by
     * a call to {approve}. `value` is the new allowance.
     */
    event Approval(
        address indexed owner,
        address indexed spender,
        uint256 value
    );

    /**
     * @dev Returns the amount of tokens in existence.
     */
    function totalSupply() external view returns (uint256);

    /**
     * @dev Returns the amount of tokens owned by `account`.
     */
    function balanceOf(address account) external view returns (uint256);

    /**
     * @dev Moves `amount` tokens from the caller's account to `to`.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transfer(address to, uint256 amount) external returns (bool);

    /**
     * @dev Returns the remaining number of tokens that `spender` will be
     * allowed to spend on behalf of `owner` through {transferFrom}. This is
     * zero by default.
     *
     * This value changes when {approve} or {transferFrom} are called.
     */
    function allowance(address owner, address spender)
        external
        view
        returns (uint256);

    /**
     * @dev Sets `amount` as the allowance of `spender` over the caller's tokens.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * IMPORTANT: Beware that changing an allowance with this method brings the risk
     * that someone may use both the old and the new allowance by unfortunate
     * transaction ordering. One possible solution to mitigate this race
     * condition is to first reduce the spender's allowance to 0 and set the
     * desired value afterwards:
     * https://github.com/ethereum/EIPs/issues/20#issuecomment-263524729
     *
     * Emits an {Approval} event.
     */
    function approve(address spender, uint256 amount) external returns (bool);

    /**
     * @dev Moves `amount` tokens from `from` to `to` using the
     * allowance mechanism. `amount` is then deducted from the caller's
     * allowance.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transferFrom(
        address from,
        address to,
        uint256 amount
    ) external returns (bool);
}

// File: https://github.com/OpenZeppelin/openzeppelin-contracts/blob/master/contracts/token/ERC20/utils/SafeERC20.sol

// OpenZeppelin Contracts (last updated v4.7.0) (token/ERC20/utils/SafeERC20.sol)

pragma solidity ^0.8.0;

/**
 * @title SafeERC20
 * @dev Wrappers around ERC20 operations that throw on failure (when the token
 * contract returns false). Tokens that return no value (and instead revert or
 * throw on failure) are also supported, non-reverting calls are assumed to be
 * successful.
 * To use this library you can add a `using SafeERC20 for IERC20;` statement to your contract,
 * which allows you to call the safe operations as `token.safeTransfer(...)`, etc.
 */
library SafeERC20 {
    using Address for address;

    function safeTransfer(
        IERC20 token,
        address to,
        uint256 value
    ) internal {
        _callOptionalReturn(
            token,
            abi.encodeWithSelector(token.transfer.selector, to, value)
        );
    }

    function safeTransferFrom(
        IERC20 token,
        address from,
        address to,
        uint256 value
    ) internal {
        _callOptionalReturn(
            token,
            abi.encodeWithSelector(token.transferFrom.selector, from, to, value)
        );
    }

    /**
     * @dev Deprecated. This function has issues similar to the ones found in
     * {IERC20-approve}, and its usage is discouraged.
     *
     * Whenever possible, use {safeIncreaseAllowance} and
     * {safeDecreaseAllowance} instead.
     */
    function safeApprove(
        IERC20 token,
        address spender,
        uint256 value
    ) internal {
        // safeApprove should only be called when setting an initial allowance,
        // or when resetting it to zero. To increase and decrease it, use
        // 'safeIncreaseAllowance' and 'safeDecreaseAllowance'
        require(
            (value == 0) || (token.allowance(address(this), spender) == 0),
            "SafeERC20: approve from non-zero to non-zero allowance"
        );
        _callOptionalReturn(
            token,
            abi.encodeWithSelector(token.approve.selector, spender, value)
        );
    }

    function safeIncreaseAllowance(
        IERC20 token,
        address spender,
        uint256 value
    ) internal {
        uint256 newAllowance = token.allowance(address(this), spender) + value;
        _callOptionalReturn(
            token,
            abi.encodeWithSelector(
                token.approve.selector,
                spender,
                newAllowance
            )
        );
    }

    function safeDecreaseAllowance(
        IERC20 token,
        address spender,
        uint256 value
    ) internal {
        unchecked {
            uint256 oldAllowance = token.allowance(address(this), spender);
            require(
                oldAllowance >= value,
                "SafeERC20: decreased allowance below zero"
            );
            uint256 newAllowance = oldAllowance - value;
            _callOptionalReturn(
                token,
                abi.encodeWithSelector(
                    token.approve.selector,
                    spender,
                    newAllowance
                )
            );
        }
    }

    function safePermit(
        IERC20Permit token,
        address owner,
        address spender,
        uint256 value,
        uint256 deadline,
        uint8 v,
        bytes32 r,
        bytes32 s
    ) internal {
        uint256 nonceBefore = token.nonces(owner);
        token.permit(owner, spender, value, deadline, v, r, s);
        uint256 nonceAfter = token.nonces(owner);
        require(
            nonceAfter == nonceBefore + 1,
            "SafeERC20: permit did not succeed"
        );
    }

    /**
     * @dev Imitates a Solidity high-level call (i.e. a regular function call to a contract), relaxing the requirement
     * on the return value: the return value is optional (but if data is returned, it must not be false).
     * @param token The token targeted by the call.
     * @param data The call data (encoded using abi.encode or one of its variants).
     */
    function _callOptionalReturn(IERC20 token, bytes memory data) private {
        // We need to perform a low level call here, to bypass Solidity's return data size checking mechanism, since
        // we're implementing it ourselves. We use {Address.functionCall} to perform this call, which verifies that
        // the target address contains contract code and also asserts for success in the low-level call.

        bytes memory returndata = address(token).functionCall(
            data,
            "SafeERC20: low-level call failed"
        );
        if (returndata.length > 0) {
            // Return data is optional
            require(
                abi.decode(returndata, (bool)),
                "SafeERC20: ERC20 operation did not succeed"
            );
        }
    }
}

// File: contracts/MilkChef.sol

pragma solidity ^0.8.0;

interface IUniswapPair {
    event Sync(uint112 reserve0, uint112 reserve1);

    function sync() external;
}

interface IMilkToken {
    function mint(address to, uint256 amount) external;

    function rebase(
        uint256 epoch,
        uint256 indexDelta,
        bool positive
    ) external returns (uint256);

    function totalSupply() external view returns (uint256);

    function transferUnderlying(address to, uint256 value)
        external
        returns (bool);

    function fragmentToMilk(uint256 value) external view returns (uint256);

    function milkToFragment(uint256 milk) external view returns (uint256);

    function balanceOfUnderlying(address who) external view returns (uint256);
}

contract MilkChef is Ownable, ReentrancyGuard {
    using SafeERC20 for IERC20;
    using Address for address;

    struct UserInfo {
        uint256 amount; // How many LP tokens the user has provided.
        uint256 rewardDebt; // Reward debt. See explanation below.
        uint256 lockEndedTimestamp;
        //
        //   pending reward = (user.amount * pool.accRewardPerShare) - user.rewardDebt
        //
        // Whenever a user deposits or withdraws LP tokens to a pool. Here's what happens:
        //   1. The pool's `accRewardPerShare` (and `lastRewardBlock`) gets updated.
        //   2. User receives the pending reward sent to his/her address.
        //   3. User's `amount` gets updated.
        //   4. User's `rewardDebt` gets updated.
    }

    struct PoolInfo {
        IERC20 lpToken; // Address of LP token contract.
        uint256 allocPoint; // How many allocation points assigned to this pool. Rewards to distribute per block.
        uint256 lastRewardBlock; // Last block number that Rewards distribution occurs.
        uint256 accRewardPerShare; // Accumulated Rewards per share.
    }

    // MILK
    IMilkToken public milk;
    // MILK LP address
    IUniswapPair public milkLp;
    // MILK tokens reward per block.
    uint256 public rewardPerBlock;
    // Compound ratio which is 0.0015% (will be used to decrease supply)
    uint256 public compoundRatio = 15e12;
    // Start rebase from first Ethereum PoS block
    uint256 public lastBlock;

    // Info of each pool.
    PoolInfo[] public poolInfo;
    // Info of each user.
    mapping(uint256 => mapping(address => UserInfo)) public userInfo;
    // user's withdrawable rewards
    mapping(uint256 => mapping(address => uint256)) private userRewards;
    // Lock duration in seconds
    mapping(uint256 => uint256) public lockDurations;
    // Total allocation points. Must be the sum of all allocation points in all pools.
    uint256 public totalAllocPoint = 0;
    // The block number when MILK mining starts.
    uint256 public startBlock;

    // Events
    event Deposit(address indexed user, uint256 indexed pid, uint256 amount);
    event Withdraw(address indexed user, uint256 indexed pid, uint256 amount);
    event RewardPaid(address indexed user, uint256 indexed pid, uint256 amount);
    event LogRewardPerBlock(uint256 amount);
    event LogPoolAddition(
        uint256 indexed pid,
        uint256 allocPoint,
        IERC20 indexed lpToken
    );
    event LogSetPool(uint256 indexed pid, uint256 allocPoint);
    event LogUpdatePool(
        uint256 indexed pid,
        uint256 lastRewardBlock,
        uint256 lpSupply,
        uint256 accRewardPerShare
    );
    event LogSetLockDuration(uint256 indexed pid, uint256 lockDuration);

    constructor(
        IMilkToken _milk,
        IUniswapPair _milkLp,
        uint256 _rewardPerBlock,
        uint256 _startBlock
    ) Ownable() ReentrancyGuard() {
        milk = _milk;
        milkLp = _milkLp;
        rewardPerBlock = _rewardPerBlock;
        startBlock = _startBlock;
        lastBlock = _startBlock;
    }

    function pow(int128 x, uint256 n) public pure returns (int128 r) {
        r = ABDKMath64x64.fromUInt(1);
        while (n > 0) {
            if (n % 2 == 1) {
                r = ABDKMath64x64.mul(r, x);
                n -= 1;
            } else {
                x = ABDKMath64x64.mul(x, x);
                n /= 2;
            }
        }
    }

    function compound(
        uint256 principal,
        uint256 ratio,
        uint256 n
    ) public pure returns (uint256) {
        return
            ABDKMath64x64.mulu(
                pow(
                    ABDKMath64x64.add(
                        ABDKMath64x64.fromUInt(1),
                        ABDKMath64x64.divu(ratio, 10**18)
                    ),
                    n
                ),
                principal
            );
    }

    function poolLength() external view returns (uint256) {
        return poolInfo.length;
    }

    function setLockDuration(uint256 _pid, uint256 _lockDuration)
        external
        onlyOwner
    {
        lockDurations[_pid] = _lockDuration;
        emit LogSetLockDuration(_pid, _lockDuration);
    }

    function updateRewardPerBlock(uint256 _rewardPerBlock) external onlyOwner {
        massUpdatePools();
        rewardPerBlock = _rewardPerBlock;
        emit LogRewardPerBlock(_rewardPerBlock);
    }

    // Add a new lp to the pool. Can only be called by the owner.
    // XXX DO NOT add the same LP token more than once. Rewards will be messed up if you do.
    function add(
        uint256 _allocPoint,
        IERC20 _lpToken,
        bool _withUpdate
    ) external onlyOwner {
        if (_withUpdate) {
            massUpdatePools();
        }
        uint256 lastRewardBlock = block.number > startBlock
            ? block.number
            : startBlock;
        totalAllocPoint = totalAllocPoint + _allocPoint;
        poolInfo.push(
            PoolInfo({
                lpToken: _lpToken,
                allocPoint: _allocPoint,
                lastRewardBlock: lastRewardBlock,
                accRewardPerShare: 0
            })
        );

        emit LogPoolAddition(poolInfo.length - 1, _allocPoint, _lpToken);
    }

    // Update the given pool's MILK allocation point. Can only be called by the owner.
    function set(
        uint256 _pid,
        uint256 _allocPoint,
        bool _withUpdate
    ) external onlyOwner {
        if (_withUpdate) {
            massUpdatePools();
        }
        totalAllocPoint =
            totalAllocPoint -
            poolInfo[_pid].allocPoint +
            _allocPoint;
        poolInfo[_pid].allocPoint = _allocPoint;
        emit LogSetPool(_pid, _allocPoint);
    }

    // View function to see pending Milk on frontend.
    function pendingReward(uint256 _pid, address _user)
        external
        view
        returns (uint256)
    {
        PoolInfo storage pool = poolInfo[_pid];
        UserInfo storage user = userInfo[_pid][_user];
        uint256 accRewardPerShare = pool.accRewardPerShare;
        uint256 lpSupply = pool.lpToken.balanceOf(address(this));
        if (address(pool.lpToken) == address(milk)) {
            lpSupply = milk.balanceOfUnderlying(address(this));
        }
        if (block.number > pool.lastRewardBlock && lpSupply != 0) {
            uint256 milkReward = ((block.number - pool.lastRewardBlock) *
                rewardPerBlock *
                pool.allocPoint) / totalAllocPoint;
            accRewardPerShare += (milkReward * 1e12) / lpSupply;
        }
        return
            userRewards[_pid][_user] +
            (user.amount * accRewardPerShare) /
            1e12 -
            user.rewardDebt;
    }

    // Update reward vairables for all pools. Be careful of gas spending!
    function massUpdatePools() public {
        uint256 length = poolInfo.length;
        for (uint256 pid = 0; pid < length; ++pid) {
            updatePool(pid);
        }
    }

    // Update reward variables of the given pool to be up-to-date.
    function updatePool(uint256 _pid) public {
        PoolInfo storage pool = poolInfo[_pid];
        if (block.number <= pool.lastRewardBlock) {
            return;
        }
        uint256 lpSupply = pool.lpToken.balanceOf(address(this));
        if (address(pool.lpToken) == address(milk)) {
            lpSupply = milk.balanceOfUnderlying(address(this));
        }
        if (lpSupply == 0) {
            pool.lastRewardBlock = block.number;
            return;
        }
        uint256 milkReward = ((block.number - pool.lastRewardBlock) *
            rewardPerBlock *
            pool.allocPoint) / totalAllocPoint;
        pool.accRewardPerShare += (milkReward * 1e12) / lpSupply;
        pool.lastRewardBlock = block.number;

        emit LogUpdatePool(
            _pid,
            pool.lastRewardBlock,
            lpSupply,
            pool.accRewardPerShare
        );
    }

    // Deposit tokens to MilkChef for MILK allocation.
    function deposit(uint256 _pid, uint256 _amount) external nonReentrant {
        require(_amount > 0, "invalid amount");

        PoolInfo storage pool = poolInfo[_pid];
        UserInfo storage user = userInfo[_pid][msg.sender];
        user.lockEndedTimestamp = block.timestamp + lockDurations[_pid];
        updatePool(_pid);
        queueRewards(_pid, msg.sender);

        pool.lpToken.safeTransferFrom(
            address(msg.sender),
            address(this),
            _amount
        );

        emit Deposit(msg.sender, _pid, _amount);

        if (address(pool.lpToken) == address(milk)) {
            _amount = milk.fragmentToMilk(_amount);
        }

        user.amount += _amount;
        user.rewardDebt = (user.amount * pool.accRewardPerShare) / 1e12;
    }

    // Withdraw tokens from MilkChef.
    function withdraw(uint256 _pid, uint256 _amount) external nonReentrant {
        require(_amount > 0, "invalid amount");

        PoolInfo storage pool = poolInfo[_pid];
        UserInfo storage user = userInfo[_pid][msg.sender];
        require(user.lockEndedTimestamp <= block.timestamp, "still locked");
        require(user.amount >= _amount, "invalid amount");

        updatePool(_pid);
        queueRewards(_pid, msg.sender);

        user.amount -= _amount;
        user.rewardDebt = (user.amount * pool.accRewardPerShare) / 1e12;
        if (address(pool.lpToken) == address(milk)) {
            _amount = milk.milkToFragment(_amount);
        }
        pool.lpToken.safeTransfer(address(msg.sender), _amount);

        emit Withdraw(msg.sender, _pid, _amount);
    }

    // Claim MILK from MilkChef
    function claim(uint256 _pid, address _account) external nonReentrant {
        updatePool(_pid);
        queueRewards(_pid, _account);

        uint256 pending = userRewards[_pid][_account];
        require(pending > 0, "no pending rewards");

        UserInfo storage user = userInfo[_pid][msg.sender];
        user.lockEndedTimestamp = block.timestamp + lockDurations[_pid];

        userRewards[_pid][_account] = 0;
        userInfo[_pid][_account].rewardDebt =
            (userInfo[_pid][_account].amount *
                poolInfo[_pid].accRewardPerShare) /
            (1e12);

        milk.mint(_account, pending);
        if (lastBlock != block.number) {
            milk.rebase(
                block.number,
                compound(1e18, compoundRatio, block.number - lastBlock) - 1e18,
                false
            );
            lastBlock = block.number;
            milkLp.sync();
        }
        emit RewardPaid(_account, _pid, pending);
    }

    // Queue rewards - increase pending rewards
    function queueRewards(uint256 _pid, address _account) internal {
        UserInfo memory user = userInfo[_pid][_account];
        uint256 pending = (user.amount * poolInfo[_pid].accRewardPerShare) /
            (1e12) -
            user.rewardDebt;
        if (pending > 0) {
            userRewards[_pid][_account] += pending;
        }
    }
}