

from itertools import count
from math import asin, pi, sqrt
def 		UpperCamelCase_(    snake_case__:						float     ) ->				float:
 return (
     (1 - 2 * point) * sqrt(point - point**2     ) + 2 * point + asin(sqrt(1 - point     )     )
 ) / 4
def 		UpperCamelCase_(    snake_case__:						int     ) ->				float:
 UpperCAmelCase__           =       (circles_number + 1 - sqrt(2 * circles_number     )) / (
     2 * (circles_number**2 + 1)
 )
 UpperCAmelCase__           =       circles_number * intersection_y

 UpperCAmelCase__           =       intersection_x * intersection_y / 2
 UpperCAmelCase__           =       circle_bottom_arc_integral(
     1 / 2     ) - circle_bottom_arc_integral(snake_case__     )

 return triangle_area + concave_region_area
def 		UpperCamelCase_(    snake_case__:						float = 1 / 10_00     ) ->				int:
 UpperCAmelCase__           =       (1 - pi / 4) / 4

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

 return -1


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






def 		UpperCamelCase_(    snake_case__:						int     ) ->				int:
 if not isinstance(snake_case__						,					snake_case__     ):
  UpperCAmelCase__           =       f"Input value of [number={number}] must be an integer"
  raise TypeError(snake_case__     )
 if number < 1:
  raise ValueError('Input must be a positive integer'     )
 return number * (2 * number - 1)


if __name__ == "__main__":
 import doctest

 doctest.testmod()



