




"""simple docstring"""






from __future__ import annotations

from collections.abc import Callable

lowercase__			=		list[list[float | int]]







def 	_snake_case     (							lowercase__				,       lowercase__						):
				_lowerCamelCase	:     int       =       len(lowercase__						)
				_lowerCamelCase	:     Matrix       =       [[0 for _ in range(size + 1						)] for _ in range(lowercase__						)]
				_lowerCamelCase	:     int
				_lowerCamelCase	:     int
				_lowerCamelCase	:     int
				_lowerCamelCase	:     int
				_lowerCamelCase	:     int
				_lowerCamelCase	:     float

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

								_lowerCamelCase	:     str       =       vector[row][0]

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

								for rowa in range(row + 1				,       lowercase__						):
												_lowerCamelCase	:     Tuple       =       augmented[rowa][col] / augmented[row][col]
												_lowerCamelCase	:     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				,       lowercase__						):
								for row in range(lowercase__						):
												_lowerCamelCase	:     Optional[int]       =       augmented[row][col] / augmented[col][col]
												for cola in range(lowercase__				,       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(lowercase__						)
				]







def 	_snake_case     (							lowercase__						):
				_lowerCamelCase	:     int       =       len(lowercase__						)
				_lowerCamelCase	:     Matrix       =       [[0 for _ in range(lowercase__						)] for _ in range(lowercase__						)]
				_lowerCamelCase	:     Matrix       =       [[0] for _ in range(lowercase__						)]
				_lowerCamelCase	:     Matrix
				_lowerCamelCase	:     int
				_lowerCamelCase	:     int
				_lowerCamelCase	:     int

				for x_val, y_val in enumerate(lowercase__						):
								for col in range(lowercase__						):
												_lowerCamelCase	:     Optional[int]       =       (x_val + 1) ** (size - col - 1)
								_lowerCamelCase	:     Dict       =       y_val

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

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

				return interpolated_func







def 	_snake_case     (							lowercase__						):
				return (
				    1
				    - variable
				    + variable**2
				    - variable**3
				    + variable**4
				    - variable**5
				    + variable**6
				    - variable**7
				    + variable**8
				    - variable**9
				    + variable**10
				)







def 	_snake_case     (							lowercase__ = question_function				,       lowercase__ = 10						):
				_lowerCamelCase	:     list[int]       =       [func(lowercase__						) for x_val in range(1				,       order + 1						)]

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

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

				for poly in polynomials:
								_lowerCamelCase	:     Tuple       =       1
								while func(lowercase__						) == poly(lowercase__						):
												x_val += 1

								ret += poly(lowercase__						)

				return ret


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





"""simple docstring"""






from __future__ import annotations







def 	_snake_case     (							lowercase__						):  # This function is recursive
				_lowerCamelCase	:     Optional[Any]       =       len(lowercase__						)
				# If the array contains only one element, we return it (it's the stop condition of
				# recursion)
				if array_length <= 1:
								return array
								# Else
				_lowerCamelCase	:     Optional[Any]       =       array[0]
				_lowerCamelCase	:     Optional[int]       =       False
				_lowerCamelCase	:     List[str]       =       1
				_lowerCamelCase	:     list[int]       =       []
				while not is_found and i < array_length:
								if array[i] < pivot:
												_lowerCamelCase	:     Union[str, Any]       =       True
												_lowerCamelCase	:     str       =       [element for element in array[i:] if element >= array[i]]
												_lowerCamelCase	:     str       =       longest_subsequence(lowercase__						)
												if len(lowercase__						) > len(lowercase__						):
																_lowerCamelCase	:     Any       =       temp_array
								else:
												i += 1

				_lowerCamelCase	:     Optional[Any]       =       [element for element in array[1:] if element >= pivot]
				_lowerCamelCase	:     Dict       =       [pivot, *longest_subsequence(lowercase__						)]
				if len(lowercase__						) > len(lowercase__						):
								return temp_array
				else:
								return longest_subseq


if __name__ == "__main__":
			import doctest

			doctest.testmod()