



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





def 		__snake_case						(    _UpperCAmelCase				):
					"""simple docstring"""






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





def 		__snake_case						(    _UpperCAmelCase				):
					"""simple docstring"""






					lowercase						   =       (circles_number + 1 - sqrt(2 * circles_number				)) / (
					    2 * (circles_number**2 + 1)
					)
					lowercase						   =       circles_number * intersection_y

					lowercase						   =       intersection_x * intersection_y / 2
					lowercase						   =       circle_bottom_arc_integral(
					    1 / 2				) - circle_bottom_arc_integral(_UpperCAmelCase				)

					return triangle_area + concave_region_area





def 		__snake_case						(    _UpperCAmelCase = 1 / 10_00				):
					"""simple docstring"""






					lowercase						   =       (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() = }""")











from __future__ import annotations





def 		__snake_case						(    _UpperCAmelCase       ,			_UpperCAmelCase       ,			_UpperCAmelCase				):
					"""simple docstring"""






					assert (
					    c % greatest_common_divisor(_UpperCAmelCase       ,			_UpperCAmelCase				) == 0
					)  # greatest_common_divisor(a,b) function implemented below
					((lowercase)   ,      (lowercase)   ,      (lowercase))						   =       extended_gcd(_UpperCAmelCase       ,			_UpperCAmelCase				)  # extended_gcd(a,b) function implemented below
					lowercase						   =       c / d
					return (r * x, r * y)





def 		__snake_case						(    _UpperCAmelCase       ,			_UpperCAmelCase       ,			_UpperCAmelCase       ,			_UpperCAmelCase = 2				):
					"""simple docstring"""






					((lowercase)   ,      (lowercase))						   =       diophantine(_UpperCAmelCase       ,			_UpperCAmelCase       ,			_UpperCAmelCase				)  # Initial value
					lowercase						   =       greatest_common_divisor(_UpperCAmelCase       ,			_UpperCAmelCase				)
					lowercase						   =       a // d
					lowercase						   =       b // d

					for i in range(_UpperCAmelCase				):
										lowercase						   =       xa + i * q
										lowercase						   =       ya - i * p
										print(_UpperCAmelCase       ,			_UpperCAmelCase				)





def 		__snake_case						(    _UpperCAmelCase       ,			_UpperCAmelCase				):
					"""simple docstring"""






					if a < b:
										lowercase   ,      lowercase						   =       b, a

					while a % b != 0:
										lowercase   ,      lowercase						   =       b, a % b

					return b





def 		__snake_case						(    _UpperCAmelCase       ,			_UpperCAmelCase				):
					"""simple docstring"""






					assert a >= 0 and b >= 0

					if b == 0:
										lowercase   ,      lowercase   ,      lowercase						   =       a, 1, 0
					else:
										((lowercase)   ,      (lowercase)   ,      (lowercase))						   =       extended_gcd(_UpperCAmelCase       ,			a % b				)
										lowercase						   =       q
										lowercase						   =       p - q * (a // b)

					assert a % d == 0 and b % d == 0
					assert d == a * x + b * y

					return (d, x, y)


if __name__ == "__main__":
		from doctest import testmod

		testmod(name='''diophantine''', verbose=True)
		testmod(name='''diophantine_all_soln''', verbose=True)
		testmod(name='''extended_gcd''', verbose=True)
		testmod(name='''greatest_common_divisor''', verbose=True)






