code: 
  
  
  
  
  
  
  from itertools import count
  from math import asin, pi, sqrt
  
  
  def 						_A     (       SCREAMING_SNAKE_CASE__						:			float ):
  				return (
  				    (1 - 2 * point) * sqrt(point - point**2 ) + 2 * point + asin(sqrt(1 - point ) )
  				) / 4
  
  
  def 						_A     (       SCREAMING_SNAKE_CASE__						:			int ):
  				UpperCamelCase :str										=     (circles_number + 1 - sqrt(2 * circles_number )) / (
  				    2 * (circles_number**2 + 1)
  				)
  				UpperCamelCase :List[Any]										=     circles_number * intersection_y
  
  				UpperCamelCase :List[str]										=     intersection_x * intersection_y / 2
  				UpperCamelCase :Optional[int]										=     circle_bottom_arc_integral(
  				    1 / 2 ) - circle_bottom_arc_integral(SCREAMING_SNAKE_CASE__ )
  
  				return triangle_area + concave_region_area
  
  
  def 						_A     (       SCREAMING_SNAKE_CASE__						:			float = 1 / 1000 ):
  				UpperCamelCase :Union[str, Any]										=     (1 - pi / 4) / 4
  
  				for n in count(1 ):
  								if concave_triangle_area(SCREAMING_SNAKE_CASE__ ) / l_section_area < fraction:
  												return n
  
  				return -1
  
  
  if __name__ == "__main__":
  	print(f'''{solution() = }''')
  
  
  
  
  
  
style_context: 
  
  
  
  
  
  
  from __future__ import annotations
  
  
  
  
  
  
  
  class     UpperCAmelCase_		:
  
  
  
        """simple docstring"""
        def __init__(       self					,       SCREAMING_SNAKE_CASE_				)       ->   None:
            UpperCamelCase :Tuple										=     value
            UpperCamelCase :Node | None										=     None
            UpperCamelCase :Node | None										=     None
  
  
  
  
  
  
  
  class     UpperCAmelCase_		:
  
  
  
        """simple docstring"""
        UpperCamelCase_       :      int
        def __init__(       self				)       ->   None:
            UpperCamelCase :List[Any]										=     0
        def 					UpperCAmelCase		(       self					,       SCREAMING_SNAKE_CASE_					,       SCREAMING_SNAKE_CASE_				)       ->   None:
            if node is None:
                return
  
            if path_sum == self.target:
                self.paths += 1
  
            if node.left:
                self.depth_first_search(node.left					,       path_sum + node.left.value				)
            if node.right:
                self.depth_first_search(node.right					,       path_sum + node.right.value				)
  
  
  
  
        def 					UpperCAmelCase		(       self					,       SCREAMING_SNAKE_CASE_					,       SCREAMING_SNAKE_CASE_ = None				)       ->   int:
            if node is None:
                return 0
            if target is not None:
                UpperCamelCase :Tuple										=     target
  
            self.depth_first_search(SCREAMING_SNAKE_CASE_					,       node.value				)
            self.path_sum(node.left				)
            self.path_sum(node.right				)
  
            return self.paths
  
  
  if __name__ == "__main__":
   import doctest
  
   doctest.testmod()
  
  
  
  
  
  