


"""simple docstring"""


from __future__ import annotations

from collections.abc import Callable

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




def 	snake_case		(  UpperCamelCase__							:    List[Any]      ,					UpperCamelCase__							:    Optional[Any]     ) ->   Dict:
      lowerCamelCase       :       int   =						len(A__     )
      lowerCamelCase       :       Matrix   =						[[0 for _ in range(size + 1     )] for _ in range(A__     )]
      lowerCamelCase       :       int
      lowerCamelCase       :       int
      lowerCamelCase       :       int
      lowerCamelCase       :       int
      lowerCamelCase       :       int
      lowerCamelCase       :       float

      for row in range(A__     ):
            for col in range(A__     ):
                  lowerCamelCase       :       Dict   =						matrix[row][col]

            lowerCamelCase       :       Union[str, Any]   =						vector[row][0]

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

            for rowa in range(row + 1      ,					A__     ):
                  lowerCamelCase       :       str   =						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      ,					A__     ):
            for row in range(A__     ):
                  lowerCamelCase       :       List[Any]   =						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 	snake_case		(  UpperCamelCase__							:    Union[str, Any]     ) ->   Optional[int]:
      lowerCamelCase       :       int   =						len(A__     )
      lowerCamelCase       :       Matrix   =						[[0 for _ in range(A__     )] for _ in range(A__     )]
      lowerCamelCase       :       Matrix   =						[[0] for _ in range(A__     )]
      lowerCamelCase       :       Matrix
      lowerCamelCase       :       int
      lowerCamelCase       :       int
      lowerCamelCase       :       int

      for x_val, y_val in enumerate(A__     ):
            for col in range(A__     ):
                  lowerCamelCase       :       int   =						(x_val + 1) ** (size - col - 1)
            lowerCamelCase       :       Optional[Any]   =						y_val

      lowerCamelCase       :       int   =						solve(A__      ,					A__     )

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

      return interpolated_func




def 	snake_case		(  UpperCamelCase__							:    Optional[Any]     ) ->   str:
      return (
          1
          - variable
          + variable**2
          - variable**3
          + variable**4
          - variable**5
          + variable**6
          - variable**7
          + variable**8
          - variable**9
          + variable**10
      )




def 	snake_case		(  UpperCamelCase__							:    Any = question_function      ,					UpperCamelCase__							:    str = 10     ) ->   int:
      lowerCamelCase       :       list[int]   =						[func(A__     ) 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       :       Optional[Any]   =						1
            while func(A__     ) == poly(A__     ):
                  x_val += 1

            ret += poly(A__     )

      return ret


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








from __future__ import annotations

lowerCAmelCase__			:      Union[str, Any]								=[
    [-1, 0],  # left
    [0, -1],  # down
    [1, 0],  # right
    [0, 1],  # up
]






def 						a__			(				A__,      A__,      A__,      A__,      A__,      ):
			SCREAMING_SNAKE_CASE_		:     List[Any]					=					[
			    [0 for col in range(len(grid[0]				)				)] for row in range(len(A__				)				)
			]  # the reference grid
			SCREAMING_SNAKE_CASE_		:     List[Any]					=					1
			SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					[
			    [0 for col in range(len(grid[0]				)				)] for row in range(len(A__				)				)
			]  # the action grid

			SCREAMING_SNAKE_CASE_		:     Optional[int]					=					init[0]
			SCREAMING_SNAKE_CASE_		:     List[str]					=					init[1]
			SCREAMING_SNAKE_CASE_		:     Dict					=					0
			SCREAMING_SNAKE_CASE_		:     List[Any]					=					g + heuristic[x][y]  # cost from starting cell to destination cell
			SCREAMING_SNAKE_CASE_		:     List[str]					=					[[f, g, x, y]]

			SCREAMING_SNAKE_CASE_		:     Dict					=					False  # flag that is set when search is complete
			SCREAMING_SNAKE_CASE_		:     Tuple					=					False  # flag set if we can't find expand

			while not found and not resign:
						if len(A__				) == 0:
									raise ValueError('Algorithm is unable to find solution'				)
						else:  # to choose the least costliest action so as to move closer to the goal
									cell.sort()
									cell.reverse()
									SCREAMING_SNAKE_CASE_		:     Any					=					cell.pop()
									SCREAMING_SNAKE_CASE_		:     Any					=					next_cell[2]
									SCREAMING_SNAKE_CASE_		:     str					=					next_cell[3]
									SCREAMING_SNAKE_CASE_		:     Tuple					=					next_cell[1]

									if x == goal[0] and y == goal[1]:
												SCREAMING_SNAKE_CASE_		:     Dict					=					True
									else:
												for i in range(len(A__				)				):  # to try out different valid actions
															SCREAMING_SNAKE_CASE_		:     Optional[int]					=					x + DIRECTIONS[i][0]
															SCREAMING_SNAKE_CASE_		:     List[Any]					=					y + DIRECTIONS[i][1]
															if xa >= 0 and xa < len(A__				) and ya >= 0 and ya < len(grid[0]				):
																		if closed[xa][ya] == 0 and grid[xa][ya] == 0:
																					SCREAMING_SNAKE_CASE_		:     List[Any]					=					g + cost
																					SCREAMING_SNAKE_CASE_		:     List[Any]					=					ga + heuristic[xa][ya]
																					cell.append([fa, ga, xa, ya]				)
																					SCREAMING_SNAKE_CASE_		:     int					=					1
																					SCREAMING_SNAKE_CASE_		:     int					=					i
			SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					[]
			SCREAMING_SNAKE_CASE_		:     List[str]					=					goal[0]
			SCREAMING_SNAKE_CASE_		:     Optional[Any]					=					goal[1]
			invpath.append([x, y]				)  # we get the reverse path from here
			while x != init[0] or y != init[1]:
						SCREAMING_SNAKE_CASE_		:     Any					=					x - DIRECTIONS[action[x][y]][0]
						SCREAMING_SNAKE_CASE_		:     int					=					y - DIRECTIONS[action[x][y]][1]
						SCREAMING_SNAKE_CASE_		:     int					=					xa
						SCREAMING_SNAKE_CASE_		:     List[Any]					=					ya
						invpath.append([x, y]				)

			SCREAMING_SNAKE_CASE_		:     List[Any]					=					[]
			for i in range(len(A__				)				):
						path.append(invpath[len(A__				) - 1 - i]				)
			return path, action


if __name__ == "__main__":
				lowerCAmelCase__			:      Union[str, Any]								=[
				    [0, 1, 0, 0, 0, 0],
				    [0, 1, 0, 0, 0, 0],  # 0 are free path whereas 1's are obstacles
				    [0, 1, 0, 0, 0, 0],
				    [0, 1, 0, 0, 1, 0],
				    [0, 0, 0, 0, 1, 0],
				]

				lowerCAmelCase__			:      Any								=[0, 0]
				# all coordinates are given in format [y,x]
				lowerCAmelCase__			:      List[str]								=[len(grid) - 1, len(grid[0]) - 1]
				lowerCAmelCase__			:      str								=1

				# the cost map which pushes the path closer to the goal
				lowerCAmelCase__			:      Union[str, Any]								=[[0 for row in range(len(grid[0]))] for col in range(len(grid))]
				for i in range(len(grid)):
								for j in range(len(grid[0])):
												lowerCAmelCase__			:      List[str]								=abs(i - goal[0]) + abs(j - goal[1])
												if grid[i][j] == 1:
																# added extra penalty in the heuristic map
																lowerCAmelCase__			:      List[str]								=99

				lowerCAmelCase__							,lowerCAmelCase__			:      Any								=search(grid, init, goal, cost, heuristic)

				print('ACTION MAP')
				for i in range(len(action)):
								print(action[i])

				for i in range(len(path)):
								print(path[i])





