


from __future__ import annotations

from collections.abc import Callable

__lowerCAmelCase		       =							list[list[float | int]]




def    snake_case_     (  snake_case			,							snake_case     )  ->				Matrix:
       lowercase__:   int  			=   len(snake_case     )
       lowercase__:   Matrix  			=   [[0 for _ in range(size + 1     )] for _ in range(snake_case     )]
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int
       lowercase__:   float

       for row in range(snake_case     ):
              for col in range(snake_case     ):
                     lowercase__:   List[Any]  			=   matrix[row][col]

              lowercase__:   Optional[int]  			=   vector[row][0]

       lowercase__:   str  			=   0
       lowercase__:   Any  			=   0
       while row < size and col < size:
              # pivoting
              lowercase__:   List[Any]  			=   max((abs(augmented[rowa][col]     ), rowa) for rowa in range(snake_case			,							snake_case     )     )[
                  1
              ]
              if augmented[pivot_row][col] == 0:
                     col += 1
                     continue
              else:
                     lowercase__			,						lowercase__:   Optional[int]  			=   augmented[pivot_row], augmented[row]

              for rowa in range(row + 1			,							snake_case     ):
                     lowercase__:   Any  			=   augmented[rowa][col] / augmented[row][col]
                     lowercase__:   int  			=   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			,							snake_case     ):
              for row in range(snake_case     ):
                     lowercase__:   Union[str, Any]  			=   augmented[row][col] / augmented[col][col]
                     for cola in range(snake_case			,							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(snake_case     )
       ]




def    snake_case_     (  snake_case     )  ->				Callable[[int], int]:
       lowercase__:   int  			=   len(snake_case     )
       lowercase__:   Matrix  			=   [[0 for _ in range(snake_case     )] for _ in range(snake_case     )]
       lowercase__:   Matrix  			=   [[0] for _ in range(snake_case     )]
       lowercase__:   Matrix
       lowercase__:   int
       lowercase__:   int
       lowercase__:   int

       for x_val, y_val in enumerate(snake_case     ):
              for col in range(snake_case     ):
                     lowercase__:   List[str]  			=   (x_val + 1) ** (size - col - 1)
              lowercase__:   str  			=   y_val

       lowercase__:   Optional[int]  			=   solve(snake_case			,							snake_case     )

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

       return interpolated_func




def    snake_case_     (  snake_case     )  ->				int:
       return (
           1
           - variable
           + variable**2
           - variable**3
           + variable**4
           - variable**5
           + variable**6
           - variable**7
           + variable**8
           - variable**9
           + variable**10
       )




def    snake_case_     (  snake_case = question_function			,							snake_case = 10     )  ->				int:
       lowercase__:   list[int]  			=   [func(snake_case     ) for x_val in range(1			,							order + 1     )]

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

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

       for poly in polynomials:
              lowercase__:   List[str]  			=   1
              while func(snake_case     ) == poly(snake_case     ):
                     x_val += 1

              ret += poly(snake_case     )

       return ret


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






def    snake_case_     (  snake_case			,							snake_case     )  ->				str:
       if number < 0 or shift_amount < 0:
              raise ValueError('both inputs must be positive integers'     )

       lowercase__:   str  			=   str(bin(snake_case     )     )
       binary_number += "0" * shift_amount
       return binary_number




def    snake_case_     (  snake_case			,							snake_case     )  ->				str:
       if number < 0 or shift_amount < 0:
              raise ValueError('both inputs must be positive integers'     )

       lowercase__:   Optional[Any]  			=   str(bin(snake_case     )     )[2:]
       if shift_amount >= len(snake_case     ):
              return "0b0"
       lowercase__:   Optional[int]  			=   binary_number[: len(snake_case     ) - shift_amount]
       return "0b" + shifted_binary_number




def    snake_case_     (  snake_case			,							snake_case     )  ->				str:
       if number >= 0:  # Get binary representation of positive number
              lowercase__:   Union[str, Any]  			=   '0' + str(bin(snake_case     )     ).strip('-'     )[2:]
       else:  # Get binary (2's complement) representation of negative number
              lowercase__:   Dict  			=   len(bin(snake_case     )[3:]     )  # Find 2's complement of number
              lowercase__:   int  			=   bin(abs(snake_case     ) - (1 << binary_number_length)     )[3:]
              lowercase__:   Any  			=   (
                  '1' + '0' * (binary_number_length - len(snake_case     )) + binary_number
              )

       if shift_amount >= len(snake_case     ):
              return "0b" + binary_number[0] * len(snake_case     )
       return (
           "0b"
           + binary_number[0] * shift_amount
           + binary_number[: len(snake_case     ) - shift_amount]
       )


if __name__ == "__main__":
 import doctest

 doctest.testmod()


