





def 		lowerCAmelCase  (	UpperCamelCase__   :   list[int]       ,			UpperCamelCase__   :   list[int]					)      ->    tuple[float, float]:

						"""simple docstring"""





						if not len(UpperCamelCase__					) == len(UpperCamelCase__					) == 3:
												raise ValueError('''Please enter a valid equation.'''					)
						if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
												raise ValueError('''Both a & b of two equations can\'t be zero.'''					)

						# Extract the coefficients
						__SCREAMING_SNAKE_CASE       ,__SCREAMING_SNAKE_CASE       ,__SCREAMING_SNAKE_CASE:				Dict           =   equationa
						__SCREAMING_SNAKE_CASE       ,__SCREAMING_SNAKE_CASE       ,__SCREAMING_SNAKE_CASE:				List[str]           =   equationa

						# Calculate the determinants of the matrices
						__SCREAMING_SNAKE_CASE:				Tuple           =   aa * ba - aa * ba
						__SCREAMING_SNAKE_CASE:				List[str]           =   ca * ba - ca * ba
						__SCREAMING_SNAKE_CASE:				Optional[Any]           =   aa * ca - aa * ca

						# Check if the system of linear equations has a solution (using Cramer's rule)
						if determinant == 0:
												if determinant_x == determinant_y == 0:
																		raise ValueError('''Infinite solutions. (Consistent system)'''					)
												else:
																		raise ValueError('''No solution. (Inconsistent system)'''					)
						else:
												if determinant_x == determinant_y == 0:
																		# Trivial solution (Inconsistent system)
																		return (0.0, 0.0)
												else:
																		__SCREAMING_SNAKE_CASE:				Optional[Any]           =   determinant_x / determinant
																		__SCREAMING_SNAKE_CASE:				Optional[Any]           =   determinant_y / determinant
																		# Non-Trivial Solution (Consistent system)
																		return (x, y)










from __future__ import annotations






def 		lowerCAmelCase  (	UpperCamelCase__   :   list[int]       ,			UpperCamelCase__   :   list[int]       ,			UpperCamelCase__   :   list[int]       ,			UpperCamelCase__   :   list[list[str]]       ,			UpperCamelCase__   :   int       ,			)      ->    None:

						"""simple docstring"""





						__SCREAMING_SNAKE_CASE:				Any           =   len(UpperCamelCase__					)

						# If row is equal to the size of the board it means there are a queen in each row in
						# the current board (possible_board)
						if row == n:
												# We convert the variable possible_board that looks like this: [1, 3, 0, 2] to
												# this: ['. Q . . ', '. . . Q ', 'Q . . . ', '. . Q . ']
												boards.append(['''. ''' * i + '''Q ''' + '''. ''' * (n - 1 - i) for i in possible_board]					)
												return

						# We iterate each column in the row to find all possible results in each row
						for col in range(UpperCamelCase__					):
												# We apply that we learned previously. First we check that in the current board
												# (possible_board) there are not other same value because if there is it means
												# that there are a collision in vertical. Then we apply the two formulas we
												# learned before:
												#
												# 45º: y - x = b or 45: row - col = b
												# 135º: y + x = b or row + col = b.
												#
												# And we verify if the results of this two formulas not exist in their variables
												# respectively.  (diagonal_right_collisions, diagonal_left_collisions)
												#
												# If any or these are True it means there is a collision so we continue to the
												# next value in the for loop.
												if (
												    col in possible_board
												    or row - col in diagonal_right_collisions
												    or row + col in diagonal_left_collisions
												):
																		continue

												# If it is False we call dfs function again and we update the inputs
												depth_first_search(
												    [*possible_board, col]       ,			[*diagonal_right_collisions, row - col]       ,			[*diagonal_left_collisions, row + col]       ,			UpperCamelCase__       ,			UpperCamelCase__       ,			)






def 		lowerCAmelCase  (	UpperCamelCase__   :   int					)      ->    None:

						"""simple docstring"""





						__SCREAMING_SNAKE_CASE:				list[list[str]]           =   []
						depth_first_search([]       ,			[]       ,			[]       ,			UpperCamelCase__       ,			UpperCamelCase__					)

						# Print all the boards
						for board in boards:
												for column in board:
																		print(UpperCamelCase__					)
												print(''''''					)

						print(len(UpperCamelCase__					)       ,			'''solutions were found.'''					)


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							n_queens_solution(4)



