

"""simple docstring"""





from itertools import count
from math import asin, pi, sqrt



def        _lowerCAmelCase      (						UpperCAmelCase     :							float				):


					'''simple docstring'''






					return (
					    (1 - 2 * point) * sqrt(point - point**2				) + 2 * point + asin(sqrt(1 - point				)				)
					) / 4



def        _lowerCAmelCase      (						UpperCAmelCase     :							int				):


					'''simple docstring'''






					UpperCamelCase__		:    List[Any]         =(circles_number + 1 - sqrt(2 * circles_number				)) / (
					    2 * (circles_number**2 + 1)
					)
					UpperCamelCase__		:    int         =circles_number * intersection_y

					UpperCamelCase__		:    Tuple         =intersection_x * intersection_y / 2
					UpperCamelCase__		:    Tuple         =circle_bottom_arc_integral(
					    1 / 2				) - circle_bottom_arc_integral(UpperCAmelCase				)

					return triangle_area + concave_region_area



def        _lowerCAmelCase      (						UpperCAmelCase     :							float = 1 / 1_000				):


					'''simple docstring'''






					UpperCamelCase__		:    Any         =(1 - pi / 4) / 4

					for n in count(1				):
										if concave_triangle_area(UpperCAmelCase				) / l_section_area < fraction:
															return n

					return -1


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





"""simple docstring"""





def        _lowerCAmelCase      (						UpperCAmelCase     :							list[list[int]]				):


					'''simple docstring'''






					UpperCamelCase__		:    Dict         =[list(UpperCAmelCase				) for row in matrix]
					if matrix and isinstance(UpperCAmelCase      ,							UpperCAmelCase				):
										if isinstance(matrix[0]      ,							UpperCAmelCase				):
															UpperCamelCase__		:    List[Any]         =0
															for row in matrix:
																				if prev_len == 0:
																									UpperCamelCase__		:    Optional[Any]         =len(UpperCAmelCase				)
																									UpperCamelCase__		:    Optional[int]         =True
																				else:
																									UpperCamelCase__		:    List[str]         =prev_len == len(UpperCAmelCase				)
										else:
															UpperCamelCase__		:    Tuple         =True
					else:
										UpperCamelCase__		:    Optional[Any]         =False

					return result



def        _lowerCAmelCase      (						UpperCAmelCase     :							list[list[int]]				):


					'''simple docstring'''






					if check_matrix(UpperCAmelCase				) and len(UpperCAmelCase				) > 0:
										UpperCamelCase__		:    Optional[int]         =[list(UpperCAmelCase				) for row in a]
										UpperCamelCase__		:    Any         =len(UpperCAmelCase				)
										if isinstance(a[0]      ,							UpperCAmelCase				):
															UpperCamelCase__		:    Tuple         =len(a[0]				)
										else:
															for dat in a:
																				print(UpperCAmelCase				)
															return

										# horizotal printing increasing
										for i in range(0      ,							UpperCAmelCase				):
															print(a[0][i]				)
										# vertical printing down
										for i in range(1      ,							UpperCAmelCase				):
															print(a[i][mat_col - 1]				)
										# horizotal printing decreasing
										if mat_row > 1:
															for i in range(mat_col - 2      ,							-1      ,							-1				):
																				print(a[mat_row - 1][i]				)
        # vertical printing up
										for i in range(mat_row - 2      ,							0      ,							-1				):
															print(a[i][0]				)
										UpperCamelCase__		:    Tuple         =[row[1 : mat_col - 1] for row in a[1 : mat_row - 1]]
										if len(UpperCAmelCase				) > 0:
															spiral_print_clockwise(UpperCAmelCase				)
										else:
															return
					else:
										print('''Not a valid matrix'''				)
										return



def        _lowerCAmelCase      (						UpperCAmelCase     :							list[list]				):


					'''simple docstring'''






					if matrix:
										return list(matrix.pop(0				)				) + spiral_traversal(list(zip(*UpperCAmelCase				)				)[::-1]				)
					else:
										return []


# driver code
if __name__ == "__main__":
				import doctest

				doctest.testmod()

				_SCREAMING_SNAKE_CASE       :  int        = [[1, 2, 3, 4], [5, 6, 7, 8], [9, 1_0, 1_1, 1_2]]
				spiral_print_clockwise(a)


