from __future__ import annotations

from collections.abc import Callable

lowercase_  : int								=					list[list[float | int]]







def       A__   (						snake_case_			:		Matrix							,					snake_case_			:		Matrix   ):
 SCREAMING_SNAKE_CASE__: int=   len(snake_case_   )
 SCREAMING_SNAKE_CASE__: Matrix=   [[0 for _ in range(size + 1   )] for _ in range(snake_case_   )]
 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(snake_case_   ):
  for col in range(snake_case_   ):
   SCREAMING_SNAKE_CASE__: Union[str, Any]=   matrix[row][col]

  SCREAMING_SNAKE_CASE__: int=   vector[row][0]

 SCREAMING_SNAKE_CASE__: int=   0
 SCREAMING_SNAKE_CASE__: Union[str, Any]=   0
 while row < size and col < size:
  # pivoting
  SCREAMING_SNAKE_CASE__: int=   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:
   SCREAMING_SNAKE_CASE__,						SCREAMING_SNAKE_CASE__: str=   augmented[pivot_row], augmented[row]

  for rowa in range(row + 1							,					snake_case_   ):
   SCREAMING_SNAKE_CASE__: Union[str, Any]=   augmented[rowa][col] / augmented[row][col]
   SCREAMING_SNAKE_CASE__: Optional[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_   ):
   SCREAMING_SNAKE_CASE__: List[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       A__   (						snake_case_			:		list[int]   ):
 SCREAMING_SNAKE_CASE__: int=   len(snake_case_   )
 SCREAMING_SNAKE_CASE__: Matrix=   [[0 for _ in range(snake_case_   )] for _ in range(snake_case_   )]
 SCREAMING_SNAKE_CASE__: Matrix=   [[0] for _ in range(snake_case_   )]
 SCREAMING_SNAKE_CASE__: Matrix
 SCREAMING_SNAKE_CASE__: int
 SCREAMING_SNAKE_CASE__: int
 SCREAMING_SNAKE_CASE__: int

 for x_val, y_val in enumerate(snake_case_   ):
  for col in range(snake_case_   ):
   SCREAMING_SNAKE_CASE__: Union[str, Any]=   (x_val + 1) ** (size - col - 1)
  SCREAMING_SNAKE_CASE__: List[Any]=   y_val

 SCREAMING_SNAKE_CASE__: int=   solve(snake_case_							,					snake_case_   )

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

 return interpolated_func







def       A__   (						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       A__   (						snake_case_			:		Callable[[int], int] = question_function							,					snake_case_			:		int = 10   ):
 SCREAMING_SNAKE_CASE__: list[int]=   [func(snake_case_   ) 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__: Optional[Any]=   1
  while func(snake_case_   ) == poly(snake_case_   ):
   x_val += 1

  ret += poly(snake_case_   )

 return ret


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


import logging
import os

from .state import PartialState
class     lowerCAmelCase_     (       logging.LoggerAdapter ):







							@staticmethod
							def  UpperCamelCase_						(   _A      :     Any		):
												_UpperCamelCase			     =     PartialState()
												return not main_process_only or (main_process_only and state.is_main_process)







							def  UpperCamelCase_						(   self      :     Union[str, Any] ,			_A      :     Optional[Any] ,			_A      :     str ,			*_A      :     int ,			**_A      :     List[Any]		):
												if PartialState._shared_state == {}:
																	raise RuntimeError(
																	    '''You must initialize the accelerate state by calling either `PartialState()` or `Accelerator()` before using the logging utility.'''		)
												_UpperCamelCase			     =     kwargs.pop('''main_process_only''' ,			_A		)
												_UpperCamelCase			     =     kwargs.pop('''in_order''' ,			_A		)

												if self.isEnabledFor(_A		):
																	if self._should_log(_A		):
																						_UpperCamelCase     ,    _UpperCamelCase			     =     self.process(_A ,			_A		)
																						self.logger.log(_A ,			_A ,			*_A ,			**_A		)

																	elif in_order:
																						_UpperCamelCase			     =     PartialState()
																						for i in range(state.num_processes		):
																											if i == state.process_index:
																																_UpperCamelCase     ,    _UpperCamelCase			     =     self.process(_A ,			_A		)
																																self.logger.log(_A ,			_A ,			*_A ,			**_A		)
																											state.wait_for_everyone()




def 					_snake_case	(  __snake_case  ,     __snake_case = None						):
					if log_level is None:
										_UpperCamelCase			     =     os.environ.get('''ACCELERATE_LOG_LEVEL'''  ,     __snake_case						)
					_UpperCamelCase			     =     logging.getLogger(__snake_case						)
					if log_level is not None:
										logger.setLevel(log_level.upper()						)
										logger.root.setLevel(log_level.upper()						)
					return MultiProcessAdapter(__snake_case  ,     {}						)
