






from __future__ import annotations

from collections.abc import Callable

SCREAMING_SNAKE_CASE__   :      Optional[Any]				=		list[list[float | int]]

def   __lowercase       (			snake_case,	snake_case			):







				"""simple docstring"""






				__magic_name__		:int   =   len(snake_case			)
				__magic_name__		:Matrix   =   [[0 for _ in range(size + 1			)] for _ in range(snake_case			)]
				__magic_name__		:int
				__magic_name__		:int
				__magic_name__		:int
				__magic_name__		:int
				__magic_name__		:int
				__magic_name__		:float

				for row in range(snake_case			):
								for col in range(snake_case			):
												__magic_name__		:Optional[Any]   =   matrix[row][col]

								__magic_name__		:Optional[Any]   =   vector[row][0]

				__magic_name__		:Tuple   =   0
				__magic_name__		:str   =   0
				while row < size and col < size:
								# pivoting
								__magic_name__		: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:
												__magic_name__  ,					__magic_name__		:List[str]   =   augmented[pivot_row], augmented[row]

								for rowa in range(row + 1,	snake_case			):
												__magic_name__		:List[Any]   =   augmented[rowa][col] / augmented[row][col]
												__magic_name__		:Union[str, Any]   =   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			):
												__magic_name__		:int   =   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],	1_0			)] for row in range(snake_case			)
				]

def   __lowercase       (			snake_case			):







				"""simple docstring"""






				__magic_name__		:int   =   len(snake_case			)
				__magic_name__		:Matrix   =   [[0 for _ in range(snake_case			)] for _ in range(snake_case			)]
				__magic_name__		:Matrix   =   [[0] for _ in range(snake_case			)]
				__magic_name__		:Matrix
				__magic_name__		:int
				__magic_name__		:int
				__magic_name__		:int

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

				__magic_name__		:List[str]   =   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   __lowercase       (			snake_case			):







				"""simple docstring"""






				return (
				    1
				    - variable
				    + variable**2
				    - variable**3
				    + variable**4
				    - variable**5
				    + variable**6
				    - variable**7
				    + variable**8
				    - variable**9
				    + variable**1_0
				)

def   __lowercase       (			snake_case = question_function,	snake_case = 1_0			):







				"""simple docstring"""






				__magic_name__		:list[int]   =   [func(snake_case			) for x_val in range(1,	order + 1			)]

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

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

				for poly in polynomials:
								__magic_name__		: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   __lowercase       (			snake_case			):







				"""simple docstring"""






				if not isinstance(snake_case,	snake_case			):
								raise ValueError('''multiplicative_persistence() only accepts integral values'''			)
				if num < 0:
								raise ValueError('''multiplicative_persistence() does not accept negative values'''			)

				__magic_name__		:str   =   0
				__magic_name__		:Dict   =   str(snake_case			)

				while len(snake_case			) != 1:
								__magic_name__		:Optional[Any]   =   [int(snake_case			) for i in num_string]

								__magic_name__		:Dict   =   1
								for i in range(0,	len(snake_case			)			):
												total *= numbers[i]

								__magic_name__		:int   =   str(snake_case			)

								steps += 1
				return steps

def   __lowercase       (			snake_case			):







				"""simple docstring"""






				if not isinstance(snake_case,	snake_case			):
								raise ValueError('''additive_persistence() only accepts integral values'''			)
				if num < 0:
								raise ValueError('''additive_persistence() does not accept negative values'''			)

				__magic_name__		:str   =   0
				__magic_name__		:Union[str, Any]   =   str(snake_case			)

				while len(snake_case			) != 1:
								__magic_name__		:str   =   [int(snake_case			) for i in num_string]

								__magic_name__		:Optional[int]   =   0
								for i in range(0,	len(snake_case			)			):
												total += numbers[i]

								__magic_name__		:int   =   str(snake_case			)

								steps += 1
				return steps


if __name__ == "__main__":
					import doctest

					doctest.testmod()






