





from __future__ import annotations

from collections.abc import Callable

SCREAMING_SNAKE_CASE_						       =	list[list[float | int]]



def        __lowercase       (				_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     )			->       Matrix:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE		=						len(__A     )
      SCREAMING_SNAKE_CASE		=						[[0 for _ in range(size + 1     )] for _ in range(__A     )]
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42

      for row in range(__A     ):
            for col in range(__A     ):
                  SCREAMING_SNAKE_CASE		=						matrix[row][col]

            SCREAMING_SNAKE_CASE		=						vector[row][0]

      SCREAMING_SNAKE_CASE		=						0
      SCREAMING_SNAKE_CASE		=						0
      while row < size and col < size:
            # pivoting
            SCREAMING_SNAKE_CASE		=						max((abs(augmented[rowa][col]     ), rowa) for rowa in range(__A			,						__A     )     )[
                1
            ]
            if augmented[pivot_row][col] == 0:
                  col += 1
                  continue
            else:
                  SCREAMING_SNAKE_CASE, SCREAMING_SNAKE_CASE		=						augmented[pivot_row], augmented[row]

            for rowa in range(row + 1			,						__A     ):
                  SCREAMING_SNAKE_CASE		=						augmented[rowa][col] / augmented[row][col]
                  SCREAMING_SNAKE_CASE		=						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			,						__A     ):
            for row in range(__A     ):
                  SCREAMING_SNAKE_CASE		=						augmented[row][col] / augmented[col][col]
                  for cola in range(__A			,						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(__A     )
      ]



def        __lowercase       (				_SCREAMING_SNAKE_CASE     )			->       Callable[[int], int]:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE		=						len(__A     )
      SCREAMING_SNAKE_CASE		=						[[0 for _ in range(__A     )] for _ in range(__A     )]
      SCREAMING_SNAKE_CASE		=						[[0] for _ in range(__A     )]
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42

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

      SCREAMING_SNAKE_CASE		=						solve(__A			,						__A     )

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

      return interpolated_func



def        __lowercase       (				_SCREAMING_SNAKE_CASE     )			->       int:


      '''simple docstring'''


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



def        __lowercase       (				_SCREAMING_SNAKE_CASE = question_function			,						_SCREAMING_SNAKE_CASE = 10     )			->       int:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE		=						[func(__A     ) for x_val in range(1			,						order + 1     )]

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

      SCREAMING_SNAKE_CASE		=						0
      SCREAMING_SNAKE_CASE		=						42
      SCREAMING_SNAKE_CASE		=						42

      for poly in polynomials:
            SCREAMING_SNAKE_CASE		=						1
            while func(__A     ) == poly(__A     ):
                  x_val += 1

            ret += poly(__A     )

      return ret


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









class 		UpperCamelCase__			:



     '''simple docstring'''

     def __init__(							self						:   int	)					->		str:




           '''simple docstring'''


           SCREAMING_SNAKE_CASE		=						{}






     def     SCREAMING_SNAKE_CASE__  (							self						:   List[Any]	)					->		None:




           '''simple docstring'''


           print(self.vertex	)
           for i in self.vertex:
                 print(lowerCamelCase__						,""" -> """						,""" -> """.join([str(lowerCamelCase__	) for j in self.vertex[i]]	)	)






     def     SCREAMING_SNAKE_CASE__  (							self						:   List[Any]						,lowerCamelCase__						:   int						,lowerCamelCase__						:   int	)					->		None:




           '''simple docstring'''


           if from_vertex in self.vertex:
                 self.vertex[from_vertex].append(lowerCamelCase__	)
           else:
                 # else make a new vertex
                 SCREAMING_SNAKE_CASE		=						[to_vertex]






     def     SCREAMING_SNAKE_CASE__  (							self						:   str	)					->		None:




           '''simple docstring'''


           SCREAMING_SNAKE_CASE		=						[False] * len(self.vertex	)

           # call the recursive helper function
           for i in range(len(self.vertex	)	):
                 if not visited[i]:
                       self.dfs_recursive(lowerCamelCase__						,lowerCamelCase__	)






     def     SCREAMING_SNAKE_CASE__  (							self						:   Any						,lowerCamelCase__						:   int						,lowerCamelCase__						:   list	)					->		None:




           '''simple docstring'''


           SCREAMING_SNAKE_CASE		=						True

           print(lowerCamelCase__						,end=""" """	)

           # Recur for all the vertices that are adjacent to this node
           for i in self.vertex:
                 if not visited[i]:
                       self.dfs_recursive(lowerCamelCase__						,lowerCamelCase__	)


if __name__ == "__main__":
      SCREAMING_SNAKE_CASE_						       =	Graph()
      g.add_edge(0, 1)
      g.add_edge(0, 2)
      g.add_edge(1, 2)
      g.add_edge(2, 0)
      g.add_edge(2, 3)
      g.add_edge(3, 3)

      g.print_graph()
      print("""DFS:""")
      g.dfs()

      # OUTPUT:
      # 0  ->  1 -> 2
      # 1  ->  2
      # 2  ->  0 -> 3
      # 3  ->  3
      # DFS:
      #  0 1 2 3


