



def     a(      A       :    str       )		->       str:
					"""simple docstring"""





					a					     =   0

					# if input_string is "aba" than new_input_string become "a|b|a"
					a					     =   ""
					a					     =   ""

					# append each character + "|" in new_string for range(0, length-1)
					for i in input_string[: len(A       ) - 1]:
										new_input_string += i + "|"
					# append last character
					new_input_string += input_string[-1]

					# we will store the starting and ending of previous furthest ending palindromic
					# substring
					a     ,				a					     =   0, 0

					# length[i] shows the length of palindromic substring with center i
					a					     =   [1 for i in range(len(A       )       )]

					# for each character in new_string find corresponding palindromic string
					a					     =   0
					for j in range(len(A       )       ):
										a					     =   1 if j > r else min(length[l + r - j] // 2 ,     r - j + 1       )
										while (
										    j - k >= 0
										    and j + k < len(A       )
										    and new_input_string[k + j] == new_input_string[j - k]
										):
															k += 1

										a					     =   2 * k - 1

										# does this string is ending after the previously explored end (that is r) ?
										# if yes the update the new r to the last index of this
										if j + k - 1 > r:
															a					     =   j - k + 1  # noqa: E741
															a					     =   j + k - 1

										# update max_length and start position
										if max_length < length[j]:
															a					     =   length[j]
															a					     =   j

    # create that string
					a					     =   new_input_string[start - max_length // 2 : start + max_length // 2 + 1]
					for i in s:
										if i != "|":
															output_string += i

					return output_string


if __name__ == "__main__":
				import doctest

				doctest.testmod()





from __future__ import annotations

_lowercase: Tuple      			=		list[list[int]]

# assigning initial values to the grid
_lowercase: Matrix      			=		[
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
_lowercase: Matrix      			=		[
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]






def     a(      A       :    Matrix ,     A       :    int ,     A       :    int ,     A       :    int       )		->       bool:
					"""simple docstring"""





					for i in range(9       ):
										if grid[row][i] == n or grid[i][column] == n:
															return False

					for i in range(3       ):
										for j in range(3       ):
															if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
																				return False

					return True






def     a(      A       :    Matrix       )		->       tuple[int, int] | None:
					"""simple docstring"""





					for i in range(9       ):
										for j in range(9       ):
															if grid[i][j] == 0:
																				return i, j
					return None






def     a(      A       :    Matrix       )		->       Matrix | None:
					"""simple docstring"""





					if location := find_empty_location(A       ):
										a     ,				a					     =   location
					else:
										# If the location is ``None``, then the grid is solved.
										return grid

					for digit in range(1 ,     10       ):
										if is_safe(A ,     A ,     A ,     A       ):
															a					     =   digit

															if sudoku(A       ) is not None:
																				return grid

															a					     =   0

					return None






def     a(      A       :    Matrix       )		->       None:
					"""simple docstring"""





					for row in grid:
										for cell in row:
															print(A ,     end=" "       )
										print()


if __name__ == "__main__":
				# make a copy of grid so that you can compare with the unmodified grid
				for example_grid in (initial_grid, no_solution):
								print("\nExample grid:\n" + "=" * 20)
								print_solution(example_grid)
								print("\nExample grid solution:")
								_lowercase: List[str]      			=		sudoku(example_grid)
								if solution is not None:
												print_solution(solution)
								else:
												print("Cannot find a solution.")
