
"""simple docstring"""

from __future__ import annotations

from collections.abc import Callable

UpperCamelCase	:				str  	= list[list[float | int]]






def 							A						(				snake_case      :Matrix				,					snake_case      :Matrix		)      ->	Matrix:
 __UpperCamelCase				  =       len(lowerCAmelCase__		)
 __UpperCamelCase				  =       [[0 for _ in range(size + 1		)] for _ in range(lowerCAmelCase__		)]
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2

 for row in range(lowerCAmelCase__		):
  for col in range(lowerCAmelCase__		):
   __UpperCamelCase				  =       matrix[row][col]

  __UpperCamelCase				  =       vector[row][0]

 __UpperCamelCase				  =       0
 __UpperCamelCase				  =       0
 while row < size and col < size:
  # pivoting
  __UpperCamelCase				  =       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				  =       augmented[pivot_row], augmented[row]

  for rowa in range(row + 1				,					lowerCAmelCase__		):
   __UpperCamelCase				  =       augmented[rowa][col] / augmented[row][col]
   __UpperCamelCase				  =       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				  =       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]				,					1_0		)] for row in range(lowerCAmelCase__		)
 ]






def 							A						(				snake_case      :list[int]		)      ->	Callable[[int], int]:
 __UpperCamelCase				  =       len(lowerCAmelCase__		)
 __UpperCamelCase				  =       [[0 for _ in range(lowerCAmelCase__		)] for _ in range(lowerCAmelCase__		)]
 __UpperCamelCase				  =       [[0] for _ in range(lowerCAmelCase__		)]
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2

 for x_val, y_val in enumerate(lowerCAmelCase__		):
  for col in range(lowerCAmelCase__		):
   __UpperCamelCase				  =       (x_val + 1) ** (size - col - 1)
  __UpperCamelCase				  =       y_val

 __UpperCamelCase				  =       solve(lowerCAmelCase__				,					lowerCAmelCase__		)

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

 return interpolated_func






def 							A						(				snake_case      :int		)      ->	int:
 return (
     1
     - variable
     + variable**2
     - variable**3
     + variable**4
     - variable**5
     + variable**6
     - variable**7
     + variable**8
     - variable**9
     + variable**1_0
 )






def 							A						(				snake_case      :Callable[[int], int] = question_function				,					snake_case      :int = 1_0		)      ->	int:
 __UpperCamelCase				  =       [func(lowerCAmelCase__		) for x_val in range(1				,					order + 1		)]

 __UpperCamelCase				  =       [
     interpolate(data_points[:max_coeff]		) for max_coeff in range(1				,					order + 1		)
 ]

 __UpperCamelCase				  =       0
 __UpperCamelCase				  =       4_2
 __UpperCamelCase				  =       4_2

 for poly in polynomials:
  __UpperCamelCase				  =       1
  while func(lowerCAmelCase__		) == poly(lowerCAmelCase__		):
   x_val += 1

  ret += poly(lowerCAmelCase__		)

 return ret


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




"""simple docstring"""






from __future__ import annotations

from typing import Any





class  _A						:

      def __init__(						self		,       __lowerCAmelCase = 6						):







            """simple docstring"""



            lowercase														=	None
            lowercase														=	None
            self.create_linked_list(__lowerCAmelCase						)

      def 				A__       (						self		,       __lowerCAmelCase						):







            """simple docstring"""



            lowercase														=	Node()
            lowercase														=	current_node
            lowercase														=	current_node
            lowercase														=	current_node
            for _ in range(1		,       __lowerCAmelCase						):
                  lowercase														=	Node()
                  lowercase														=	current_node
                  lowercase														=	previous_node
                  lowercase														=	current_node
            lowercase														=	self.front
            lowercase														=	previous_node

      def 				A__       (						self						):







            """simple docstring"""



            return (
                self.front == self.rear
                and self.front is not None
                and self.front.data is None
            )

      def 				A__       (						self						):







            """simple docstring"""



            self.check_can_perform_operation()
            return self.front.data if self.front else None

      def 				A__       (						self		,       __lowerCAmelCase						):







            """simple docstring"""



            if self.rear is None:
                  return

            self.check_is_full()
            if not self.is_empty():
                  lowercase														=	self.rear.next
            if self.rear:
                  lowercase														=	data

      def 				A__       (						self						):







            """simple docstring"""



            self.check_can_perform_operation()
            if self.rear is None or self.front is None:
                  return None
            if self.front == self.rear:
                  lowercase														=	self.front.data
                  lowercase														=	None
                  return data

            lowercase														=	self.front
            lowercase														=	old_front.next
            lowercase														=	old_front.data
            lowercase														=	None
            return data

      def 				A__       (						self						):







            """simple docstring"""



            if self.is_empty():
                  raise Exception("""Empty Queue"""						)







      def 				A__       (						self						):







            """simple docstring"""



            if self.rear and self.rear.next == self.front:
                  raise Exception("""Full Queue"""						)






class  _A						:

      def __init__(						self						):







            """simple docstring"""



            lowercase														=	None
            lowercase														=	None
            lowercase														=	None


if __name__ == "__main__":
       import doctest

       doctest.testmod()



