




'''simple docstring'''






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

def       __lowerCAmelCase (   snake_case__						):
			return (
			    (1 - 2 * point) * sqrt(point - point**2						) + 2 * point + asin(sqrt(1 - point						)						)
			) / 4

def       __lowerCAmelCase (   snake_case__						):
			__UpperCamelCase				:      Optional[Any]									=       (circles_number + 1 - sqrt(2 * circles_number						)) / (
			    2 * (circles_number**2 + 1)
			)
			__UpperCamelCase				:      Tuple									=       circles_number * intersection_y

			__UpperCamelCase				:      Any									=       intersection_x * intersection_y / 2
			__UpperCamelCase				:      Dict									=       circle_bottom_arc_integral(
			    1 / 2						) - circle_bottom_arc_integral(snake_case__						)

			return triangle_area + concave_region_area

def       __lowerCAmelCase (   snake_case__ = 1 / 1_000						):
			__UpperCamelCase				:      str									=       (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() = }')









'''simple docstring'''






from __future__ import annotations

import random
from collections.abc import Iterable
from typing import Any, Generic, TypeVar

_lowerCAmelCase				     = TypeVar('''T''', bound=bool)



class  A		(			Generic[T]		):




				'''simple docstring'''





				def __init__(self			,  _UpperCAmelCase      )     ->	None:
							__UpperCamelCase				:      T									=       value
							__UpperCamelCase				:      RandomizedHeapNode[T] | None									=       None
							__UpperCamelCase				:      RandomizedHeapNode[T] | None									=       None
				@property
				def 	a_	(self      )     ->	T:
							return self._value





				@staticmethod
				def 	a_	(_UpperCAmelCase			,  _UpperCAmelCase      )     ->	RandomizedHeapNode[T] | None:
							if not roota:
										return roota

							if not roota:
										return roota

							if roota.value > roota.value:
										__UpperCamelCase		, __UpperCamelCase				:      int									=       roota, roota

							if random.choice([True, False]      ):
										__UpperCamelCase		, __UpperCamelCase				:      Dict									=       roota.right, roota.left

							__UpperCamelCase				:      Any									=       RandomizedHeapNode.merge(roota.left			,  _UpperCAmelCase      )

							return roota







class  A		(			Generic[T]		):




				'''simple docstring'''





				def __init__(self			,  _UpperCAmelCase = ()      )     ->	None:
							__UpperCamelCase				:      RandomizedHeapNode[T] | None									=       None

							if data:
										for item in data:
													self.insert(_UpperCAmelCase      )
				def 	a_	(self			,  _UpperCAmelCase      )     ->	None:
							__UpperCamelCase				:      str									=       RandomizedHeapNode.merge(self._root			,  RandomizedHeapNode(_UpperCAmelCase      )      )
				def 	a_	(self      )     ->	T | None:
							__UpperCamelCase				:      Dict									=       self.top()

							if self._root is None:
										return None

							__UpperCamelCase				:      List[str]									=       RandomizedHeapNode.merge(self._root.left			,  self._root.right      )

							return result
				def 	a_	(self      )     ->	T:
							if not self._root:
										raise IndexError("Can't get top element for the empty heap."      )
							return self._root.value
				def 	a_	(self      )     ->	None:
							__UpperCamelCase				:      int									=       None
				def 	a_	(self      )     ->	list[Any]:
							__UpperCamelCase				:      int									=       []
							while self:
										result.append(self.pop()      )

							return result





				def __bool__(self      )     ->	bool:
							return self._root is not None


if __name__ == "__main__":
				import doctest

				doctest.testmod()



