// File: contracts/SplytStaking.sol



// SPDX-License-Identifier: MIT
pragma solidity 0.8.0;

import "@openzeppelin/contracts/token/ERC20/utils/SafeERC20.sol";
import "@openzeppelin/contracts/token/ERC20/IERC20.sol";
import "@openzeppelin/contracts/utils/math/SafeMath.sol";
import "@openzeppelin/contracts/access/Ownable.sol";
import "./ABDKMath64x64.sol";

/**
 * @title SplytStaking
 * @dev Enables a user to stake tokens into this smart contract
 */
contract SplytStaking is Ownable {
  using SafeMath for uint256;
  using SafeERC20 for IERC20;

  IERC20 private _token; // The token to be used in this smart contract

  struct Compound {
    uint depositStartBlock;  // Block where users are allowed to start deposits
    uint depositEndBlock;    // Block where deposits are no longer allowed
    uint interestStartBlock; // Block that starts yield gaining
    uint interestEndBlock;   // Block that ends yield gained
    uint startTime;          // Time to start computing compound yield
    uint endTime;            // Time to stop computing compound yield
    uint apyPerSec;          // Interval at which yield compounds, e.g., 31556952 seconds/year, 8600 hours/year
    uint apyPerBlock;        // Reward per block. We assume 15s per block
  }

  Compound private compound;               // Struct to hold compound information
  uint constant secondsPerYear = 31556952; // Seconds in a year
  uint constant blocksPerMonth = 175200;   // Blocks per month and max amount of blocks allowed for reward
  uint constant secondPerBlock = 15;       // Number of seconds between blocks mined
  uint totalPrincipal;                     // Total amount of token principal deposited into this smart contract
  uint numUsersWithDeposits;               // Total number of users who have deposited tokens into this smart contract
  uint feeDivisor = 10**3;                 // Withdrawal fee (0.1%)

  mapping (address => uint256) public _balances;  // Balance of tokens by user

  /**
   * @dev Emitted when `_amount` tokens are staked by `_userAddress`
   */
  event TokenStaked(address _userAddress, uint256 _amount);

  /**
   * @dev Emitted when `_amount` tokens are withdrawn by `_userAddress`
   */
  event TokenWithdrawn(address _userAddress, uint256 _amount);

  /**
   * @dev Creates the SplytStaking smart contract
   * @param token address of the token to be vested
   * @param _apyPerSec APY gained per second ratio
   * @param _depositStartBlock Block where deposits start
   * @param _depositEndBlock Block where deposits end
   */
  constructor (address token, uint _apyPerSec, uint _depositStartBlock, uint _depositEndBlock) public {
    _token = IERC20(token);

    // Compute start and end blocks for yield compounding
    uint interestStartBlock = _depositEndBlock.add(1);
    uint interestEndBlock = interestStartBlock.add(blocksPerMonth);

    // Compute start and end times for the same time period
    uint blocksUntilInterestStarts = interestStartBlock.sub(block.number);
    uint interestStartTime = block.timestamp.add(blocksUntilInterestStarts.mul(secondPerBlock));
    uint blocksUntilInterestEnd = interestEndBlock.sub(interestStartBlock);
    uint interestEndTime = block.timestamp.add(blocksUntilInterestEnd.mul(secondPerBlock));

    compound = Compound({
      depositStartBlock: _depositStartBlock,
      depositEndBlock: _depositEndBlock,
      interestStartBlock: interestStartBlock,
      interestEndBlock: interestEndBlock,
      startTime: interestStartTime,
      endTime: interestEndTime,
      apyPerSec: _apyPerSec,
      apyPerBlock: _apyPerSec.mul(secondPerBlock)
    });
  }

  // -----------------------------------------------------------------------
  // Modifiers
  // -----------------------------------------------------------------------

  /**
   * @dev Modifier that only lets the function continue if it is within the deposit window (depositStartBlock, depositEndBlock)
   */
  modifier depositWindow() {
    require (block.number > compound.depositStartBlock && block.number <= compound.depositEndBlock, "DepositWindow: Can be called only during deposit window");
    _;
  }

  // -----------------------------------------------------------------------
  // Helpers
  // -----------------------------------------------------------------------

  /**
   * @dev Check that the amount of blocks staked does not exceed limit: blocksPerMonth=maxBlock.
   * @dev If a user keeps their tokens longer than maxBlock, they only get yield up to maxBlock.
   * @param numBlocksPassed Amount of blocks passed
   * @param maxBlock Highest amount of blocks allowed to pass (limit)
   */
  function _blocksStaked(uint numBlocksPassed, uint maxBlock) public pure returns (uint) {
    if(numBlocksPassed >= maxBlock) {
      return maxBlock;
    } else {
      return numBlocksPassed;
    }
  }

  // -----------------------------------------------------------------------
  // COMPUTATIONS
  // -----------------------------------------------------------------------

  /**
   * @dev Estimates the power computation
   * @param x base
   * @param n duration
   */
  function pow (int128 x, uint 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;
    }
  }
  }

  /**
   * @dev Method to compute APY gained. Note: this is only used for comparison, not actually used to compute real gain
   * @param principal Principal provided
   * @param ratio Amount of APY gained per second
   * @param n Duration
   */
  function compoundInterestByTime(uint principal, uint ratio, uint n) public pure returns (uint) {
    return ABDKMath64x64.mulu (
      pow (
        ABDKMath64x64.add (ABDKMath64x64.fromUInt (1), ABDKMath64x64.divu (ratio, 10**18)),
        n),
      principal);
  }

  /**
   * @dev Wrapper method that is bound to the smart contract apyPerSec variable
   * @dev Enables durations to be set manually
   * @dev Not used to compute real gain
   * @param principal Principal provided
   * @param duration Duration
   * @return the amount gained: principal + yield
   */
  function compoundWithPrincipalAndTime(uint principal, uint duration) external view returns (uint) {
    return compoundInterestByTime(principal, compound.apyPerSec, duration);
  }

  /**
   * @dev Wrapper method that is bound to the smart contract apyPerSec and computes duration against live blockchain data
   * @dev Not used to compute real gain
   * @param principal Principal provided
   * @return the amount gained: principal + yield
   */
  function compoundWithPrincipal(uint principal) public view returns (uint) {
    uint duration = block.timestamp - compound.startTime;
    return compoundInterestByTime(principal, compound.apyPerSec, duration);
  }

  /**
   * @dev Wrapper method that computes gain using the callers information
   * @dev Uses all predefined variables in the smart contract and blockchain state
   * @dev Not used to compute real gain
   * @return the amount gained: principal + yield
   */
  function compoundWithPrincipalByUser() external view returns (uint) {
    return compoundWithPrincipal(_balances[msg.sender]);
  }

  /**
   * @dev Raw method that computes gain using blocks instead of time
   * @param principal Principal provided
   * @param blocksStaked Number of blocks with which to compute gain
   * @return the amount gained: principal + yield
   */
  function _compoundInterestByBlock(uint principal, uint blocksStaked) public view returns (uint) {
    uint reward = SafeMath.div(compound.apyPerBlock.mul(principal).mul(blocksStaked), 10**18);
    return reward.add(principal);
  }

  /**
   * @dev Computes yield gained using block raw function
   * @dev Makes sure that a user cannot gain more yield than blocksPerMonth as defined in the smart contract
   * @param principal Principal
   * @return the amount gained: principal + yield
   */
  function compoundInterestByBlock(uint principal) public view returns (uint) {
    uint numBlocksPassed = block.number.sub(compound.interestStartBlock);
    uint blocksStaked = _blocksStaked(numBlocksPassed, blocksPerMonth);
    return _compoundInterestByBlock(principal, blocksStaked);
  }

  // -----------------------------------------------------------------------
  // GETTERS
  // -----------------------------------------------------------------------

  /**
   * @dev Gets block and time information out of the smart contract
   * @return _currentBlock Current block on the blockchain
   * @return _depositStartBlock Block where deposits are allowed
   * @return _depositEndBlock Block where deposits are no longer allowed
   * @return _interestStartBlock Block where yield starts growing
   * @return _interestEndBlock Block where yield stops growing
   * @return _interestStartTime Estimated yield start time (for comparison only)
   * @return _interestEndTime Estimated yield end time (for comparison only)
   * @return _interestApyPerSec APY per second rate defined in the smart contract
   * @return _interestApyPerBlock APY per block defined in the smart contract
   */
  function getCompoundInfo() external view returns (
    uint _currentBlock,
    uint _depositStartBlock,
    uint _depositEndBlock,
    uint _interestStartBlock,
    uint _interestEndBlock,
    uint _interestStartTime,
    uint _interestEndTime,
    uint _interestApyPerSec,
    uint _interestApyPerBlock
  ) {
    return (
    block.number,
    compound.depositStartBlock,
    compound.depositEndBlock,
    compound.interestStartBlock,
    compound.interestEndBlock,
    compound.startTime,
    compound.endTime,
    compound.apyPerSec,
    compound.apyPerBlock
    );
  }

  /**
   * @dev Gets staking data from this smart contract
   * @return _totalPrincipal Total amount of tokens deposited as principal
   * @return _numUsersWithDeposits Number of users who have staked into this smart contract
   */
  function getAdminStakingInfo() public view returns (uint _totalPrincipal, uint _numUsersWithDeposits) {
    return (totalPrincipal, numUsersWithDeposits);
  }

  /**
   * @dev Gets user balance data
   * @dev If this is called before any yield is gained, we manually display 0 reward
   * @param userAddress Address of a given user
   * @return _principal Principal that a user has staked
   * @return _reward Current estimated reward earned
   * @return _balance Total balance (_principal + _reward)
   */
  function getUserBalances(address userAddress) external view returns (uint _principal, uint _reward, uint _balance) {
    if(block.number <= compound.interestStartBlock) {
      return (
      _balances[userAddress],
      0,
      _balances[userAddress]);
    } else {
      uint totalBalance = compoundInterestByBlock(_balances[userAddress]);
      uint reward = totalBalance.sub(_balances[userAddress]);
      return (
      _balances[userAddress],
      reward,
      totalBalance
      );
    }
  }

  /**
   * @dev Reads the APY defined in the smart contract as a percentage
   * @return _apy APY in percentage form
   */
  function apy() external view returns (uint _apy) {
    return secondsPerYear * compound.apyPerSec * 100;
  }

  // -----------------------------------------------------------------------
  // SETTERS
  // -----------------------------------------------------------------------

  /**
   * @dev Enables the deposit window to be changed. Only the smart contract owner is allowed to do this
   * @dev Because blocks are not always found at the same rate, we may need to change the deposit window so events start on time
   * @dev We will only call this so the start time is as accurate as possible
   * @dev We have to recompute the yield start block and yield end block times as well
   * @param _depositStartBlock New deposit start block
   * @param _depositEndBlock New deposit end block
   */
  function changeDepositWindow(uint _depositStartBlock, uint _depositEndBlock) external onlyOwner {
    compound.depositStartBlock = _depositStartBlock;
    compound.depositEndBlock = _depositEndBlock;
    compound.interestStartBlock = _depositEndBlock.add(1);
    compound.interestEndBlock = compound.interestStartBlock.add(blocksPerMonth);

    uint blocksUntilInterestStarts = compound.interestStartBlock.sub(block.number);
    compound.startTime = block.timestamp.add(blocksUntilInterestStarts.mul(secondPerBlock));

    uint blocksUntilInterestEnd = compound.interestEndBlock.sub(compound.interestStartBlock);
    compound.endTime = block.timestamp.add(blocksUntilInterestEnd.mul(secondPerBlock));
  }

  /**
   * @dev Enables a user to deposit their stake into this smart contract. A user must call approve tokens before calling this method
   * @dev This can only be called during the deposit window. Calling this before or after will fail
   * @dev We also make sure to track the state of [totalPrincipal, numUsersWithDeposits]
   * @param _amount The amount of tokens to stake into this smart contract
   */
  function stakeTokens(uint _amount) external depositWindow {
    require(_token.allowance(msg.sender, address(this)) >= _amount, "TokenBalance: User has not allowed tokens to be used");
    require(_token.balanceOf(msg.sender) >= _amount, "TokenBalance: msg.sender can not stake more than they have");

    if(_balances[msg.sender] == 0) {
      numUsersWithDeposits += 1;
    }
    _balances[msg.sender] += _amount;
    totalPrincipal += _amount;

    require(_token.transferFrom(msg.sender,address(this), _amount), "TokenTransfer: failed to transfer tokens from msg.sender here");
    emit TokenStaked(msg.sender, _amount);
  }

  /**
   * @dev Lets a user withdraw all their tokens from this smart contract
   * @dev A fee is charged on all withdrawals
   * @dev We make sure to track the state of [totalPrincipal, numUsersWithDeposits]
   */
  function withdrawTokens() external {
    require(_balances[msg.sender] > 0, "TokenBalance: no tokens available to be withdrawn");

    uint totalBalance = 0;

    if(block.number <= compound.depositEndBlock) {
      totalBalance = _balances[msg.sender];
    } else {
      totalBalance = compoundInterestByBlock(_balances[msg.sender]);
    }
    uint fee = totalBalance.div(feeDivisor);
    totalBalance = totalBalance.sub(fee);

    totalPrincipal -= _balances[msg.sender];
    _balances[msg.sender] = 0;
    numUsersWithDeposits -= 1;

    require(_token.transfer(msg.sender, totalBalance));
    emit TokenWithdrawn(msg.sender, totalBalance);
  }

  /**
   * @dev Computes the amount of tokens the owner is allowed to withdraw
   * @dev Assumes owner deposited tokens at the end of the deposit window, and not all users stay for the full 30 days
   * @dev There will be a remainder because users leave before the 30 days is over. Owner withdraws the balance
   */
  function adminWithdrawRemaining() external onlyOwner {
    uint totalBalanceNeeded = compoundInterestByBlock(totalPrincipal);
    uint contractBalance = _token.balanceOf(address(this));

    // We deposit tokens and assume everyone gains yield for the full 30 days
    // There is a difference because some users will withdraw tokens before the 30 days is over
    if(contractBalance > totalBalanceNeeded) {
      uint extraTokenBalance = contractBalance.sub(totalBalanceNeeded);
      require(_token.transfer(msg.sender, extraTokenBalance));
      emit TokenWithdrawn(msg.sender, extraTokenBalance);
    }
  }

  /**
   * @dev Lets owner, and only the owner, withdraw any amount of tokens
   * @dev ONLY TO BE CALLED IN A CATASTROPHIC SCENARIO!!!
   * @param _amount Number of tokens to withdraw
   */
  function withdrawExtraTokens(uint _amount) external onlyOwner {
    require(_token.transfer(msg.sender, _amount));
    emit TokenWithdrawn(msg.sender, _amount);
  }
}

// File: @openzeppelin/contracts/token/ERC20/utils/SafeERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../IERC20.sol";
import "../../../utils/Address.sol";

/**
 * @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));
        }
    }

    /**
     * @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: @openzeppelin/contracts/token/ERC20/IERC20.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @dev Interface of the ERC20 standard as defined in the EIP.
 */
interface IERC20 {
    /**
     * @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 `recipient`.
     *
     * Returns a boolean value indicating whether the operation succeeded.
     *
     * Emits a {Transfer} event.
     */
    function transfer(address recipient, 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 `sender` to `recipient` 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 sender,
        address recipient,
        uint256 amount
    ) external returns (bool);

    /**
     * @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);
}

// File: @openzeppelin/contracts/utils/math/SafeMath.sol



// SPDX-License-Identifier: MIT

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 no longer needed starting with Solidity 0.8. 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 substraction 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: @openzeppelin/contracts/access/Ownable.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

import "../utils/Context.sol";

/**
 * @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() {
        _setOwner(_msgSender());
    }

    /**
     * @dev Returns the address of the current owner.
     */
    function owner() public view virtual returns (address) {
        return _owner;
    }

    /**
     * @dev Throws if called by any account other than the owner.
     */
    modifier onlyOwner() {
        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 {
        _setOwner(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");
        _setOwner(newOwner);
    }

    function _setOwner(address newOwner) private {
        address oldOwner = _owner;
        _owner = newOwner;
        emit OwnershipTransferred(oldOwner, newOwner);
    }
}

// File: contracts/ABDKMath64x64.sol



// SPDX-License-Identifier: BSD-4-Clause
/*
 * 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: @openzeppelin/contracts/utils/Address.sol



// SPDX-License-Identifier: MIT

pragma solidity ^0.8.0;

/**
 * @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
     * ====
     */
    function isContract(address account) internal view returns (bool) {
        // This method relies on extcodesize, which returns 0 for contracts in
        // construction, since the code is only stored at the end of the
        // constructor execution.

        uint256 size;
        assembly {
            size := extcodesize(account)
        }
        return size > 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 functionCall(target, data, "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");
        require(isContract(target), "Address: call to non-contract");

        (bool success, bytes memory returndata) = target.call{value: value}(data);
        return _verifyCallResult(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) {
        require(isContract(target), "Address: static call to non-contract");

        (bool success, bytes memory returndata) = target.staticcall(data);
        return _verifyCallResult(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) {
        require(isContract(target), "Address: delegate call to non-contract");

        (bool success, bytes memory returndata) = target.delegatecall(data);
        return _verifyCallResult(success, returndata, errorMessage);
    }

    function _verifyCallResult(
        bool success,
        bytes memory returndata,
        string memory errorMessage
    ) private pure returns (bytes memory) {
        if (success) {
            return returndata;
        } else {
            // 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

                assembly {
                    let returndata_size := mload(returndata)
                    revert(add(32, returndata), returndata_size)
                }
            } else {
                revert(errorMessage);
            }
        }
    }
}

// File: @openzeppelin/contracts/utils/Context.sol



// SPDX-License-Identifier: MIT

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;
    }
}