




from __future__ import annotations

from collections.abc import Callable

UpperCAmelCase__ :     str				    =    list[list[float | int]]






def 					__lowercase      (     _A						,    _A			)     ->						str:
      SCREAMING_SNAKE_CASE :      int       =   len(_A			)
      SCREAMING_SNAKE_CASE :      Matrix       =   [[0 for _ in range(size + 1			)] for _ in range(_A			)]
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      float

      for row in range(_A			):
            for col in range(_A			):
                  SCREAMING_SNAKE_CASE :      List[Any]       =   matrix[row][col]

            SCREAMING_SNAKE_CASE :      Union[str, Any]       =   vector[row][0]

      SCREAMING_SNAKE_CASE :      str       =   0
      SCREAMING_SNAKE_CASE :      Dict       =   0
      while row < size and col < size:
            # pivoting
            SCREAMING_SNAKE_CASE :      Tuple       =   max((abs(augmented[rowa][col]			), rowa) for rowa in range(_A						,    _A			)			)[
                1
            ]
            if augmented[pivot_row][col] == 0:
                  col += 1
                  continue
            else:
                  SCREAMING_SNAKE_CASE :      List[str]       =   augmented[pivot_row], augmented[row]

            for rowa in range(row + 1						,    _A			):
                  SCREAMING_SNAKE_CASE :      List[str]       =   augmented[rowa][col] / augmented[row][col]
                  SCREAMING_SNAKE_CASE :      str       =   0
                  for cola in range(col + 1						,    size + 1			):
                        augmented[rowa][cola] -= augmented[row][cola] * ratio

            row += 1
            col += 1

      # back substitution
      for col in range(1						,    _A			):
            for row in range(_A			):
                  SCREAMING_SNAKE_CASE :      Union[str, Any]       =   augmented[row][col] / augmented[col][col]
                  for cola in range(_A						,    size + 1			):
                        augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
      return [
          [round(augmented[row][size] / augmented[row][row]						,    10			)] for row in range(_A			)
      ]






def 					__lowercase      (     _A			)     ->						Dict:
      SCREAMING_SNAKE_CASE :      int       =   len(_A			)
      SCREAMING_SNAKE_CASE :      Matrix       =   [[0 for _ in range(_A			)] for _ in range(_A			)]
      SCREAMING_SNAKE_CASE :      Matrix       =   [[0] for _ in range(_A			)]
      SCREAMING_SNAKE_CASE :      Matrix
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      int
      SCREAMING_SNAKE_CASE :      int

      for x_val, y_val in enumerate(_A			):
            for col in range(_A			):
                  SCREAMING_SNAKE_CASE :      str       =   (x_val + 1) ** (size - col - 1)
            SCREAMING_SNAKE_CASE :      Tuple       =   y_val

      SCREAMING_SNAKE_CASE :      Tuple       =   solve(_A						,    _A			)

      def interpolated_func(_A			) -> int:
            return sum(
                round(coeffs[x_val][0]			) * (var ** (size - x_val - 1))
                for x_val in range(_A			)			)

      return interpolated_func






def 					__lowercase      (     _A			)     ->						Union[str, Any]:
      return (
          1
          - variable
          + variable**2
          - variable**3
          + variable**4
          - variable**5
          + variable**6
          - variable**7
          + variable**8
          - variable**9
          + variable**10
      )






def 					__lowercase      (     _A = question_function						,    _A = 10			)     ->						Dict:
      SCREAMING_SNAKE_CASE :      list[int]       =   [func(_A			) for x_val in range(1						,    order + 1			)]

      SCREAMING_SNAKE_CASE :      list[Callable[[int], int]]       =   [
          interpolate(data_points[:max_coeff]			) for max_coeff in range(1						,    order + 1			)
      ]

      SCREAMING_SNAKE_CASE :      int       =   0
      SCREAMING_SNAKE_CASE :      Callable[[int], int]
      SCREAMING_SNAKE_CASE :      int

      for poly in polynomials:
            SCREAMING_SNAKE_CASE :      int       =   1
            while func(_A			) == poly(_A			):
                  x_val += 1

            ret += poly(_A			)

      return ret


if __name__ == "__main__":
  print(F"""{solution() = }""")







def 					__lowercase      (     _A = 50			)     ->						int:
      SCREAMING_SNAKE_CASE :      Tuple       =   [1] * (length + 1)

      for row_length in range(3						,    length + 1			):
            for block_length in range(3						,    row_length + 1			):
                  for block_start in range(row_length - block_length			):
                        ways_number[row_length] += ways_number[
                            row_length - block_start - block_length - 1
                        ]

                  ways_number[row_length] += 1

      return ways_number[length]


if __name__ == "__main__":
  print(F"""{solution() = }""")




