


'''simple docstring'''
from __future__ import annotations

from collections.abc import Callable

UpperCamelCase__    					=      list[list[float | int]]





def 					a__       (					lowerCAmelCase__  ,					lowerCAmelCase__						)    ->	Matrix:
    UpperCAmelCase__			:		int													=				len(lowerCAmelCase__						)
    UpperCAmelCase__			:		Matrix													=				[[0 for _ in range(size + 1						)] for _ in range(lowerCAmelCase__						)]
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		float

    for row in range(lowerCAmelCase__						):
        for col in range(lowerCAmelCase__						):
            UpperCAmelCase__			:		Optional[Any]													=				matrix[row][col]

        UpperCAmelCase__			:		Tuple													=				vector[row][0]

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

        for rowa in range(row + 1  ,					lowerCAmelCase__						):
            UpperCAmelCase__			:		List[Any]													=				augmented[rowa][col] / augmented[row][col]
            UpperCAmelCase__			:		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  ,					lowerCAmelCase__						):
        for row in range(lowerCAmelCase__						):
            UpperCAmelCase__			:		int													=				augmented[row][col] / augmented[col][col]
            for cola in range(lowerCAmelCase__  ,					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(lowerCAmelCase__						)
    ]





def 					a__       (					lowerCAmelCase__						)    ->	Callable[[int], int]:
    UpperCAmelCase__			:		int													=				len(lowerCAmelCase__						)
    UpperCAmelCase__			:		Matrix													=				[[0 for _ in range(lowerCAmelCase__						)] for _ in range(lowerCAmelCase__						)]
    UpperCAmelCase__			:		Matrix													=				[[0] for _ in range(lowerCAmelCase__						)]
    UpperCAmelCase__			:		Matrix
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		int
    UpperCAmelCase__			:		int

    for x_val, y_val in enumerate(lowerCAmelCase__						):
        for col in range(lowerCAmelCase__						):
            UpperCAmelCase__			:		Optional[Any]													=				(x_val + 1) ** (size - col - 1)
        UpperCAmelCase__			:		Union[str, Any]													=				y_val

    UpperCAmelCase__			:		Any													=				solve(lowerCAmelCase__  ,					lowerCAmelCase__						)

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

    return interpolated_func





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





def 					a__       (					lowerCAmelCase__ = question_function  ,					lowerCAmelCase__ = 10						)    ->	int:
    UpperCAmelCase__			:		list[int]													=				[func(lowerCAmelCase__						) for x_val in range(1  ,					order + 1						)]

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

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

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

        ret += poly(lowerCAmelCase__						)

    return ret


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




'''simple docstring'''
def 					a__       (					lowerCAmelCase__						)    ->	int:
    if a < 0:
        raise ValueError('''Input value must be a positive integer'''						)
    elif isinstance(lowerCAmelCase__  ,					lowerCAmelCase__						):
        raise TypeError('''Input value must be a \'int\' type'''						)
    return bin(lowerCAmelCase__						).count('''1'''						)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
