'''simple docstring'''





class   UpperCamelCase					:







    """simple docstring"""





    class   UpperCamelCase					:







        """simple docstring"""





        A     :    int				=			"_prev", "_data", "_next"





        def __init__(					self     :						Tuple							,							UpperCAmelCase_     :						Union[str, Any]							,							UpperCAmelCase_     :						Union[str, Any]							,							UpperCAmelCase_     :						Optional[Any]):


             """simple docstring"""







             a					:							int							= link_p
             a					:							Optional[Any]							= element
             a					:							Dict							= link_n







        def    SCREAMING_SNAKE_CASE_     (					self     :						Union[str, Any]):


             """simple docstring"""







             return (
                 f""" Prev -> {self._prev is not None}, Next -> {self._next is not None}"""
             )






    def __init__(					self     :						Any):


         """simple docstring"""







         a					:							Dict							= self._Node(UpperCAmelCase_							,							UpperCAmelCase_							,							UpperCAmelCase_)
         a					:							int							= self._Node(UpperCAmelCase_							,							UpperCAmelCase_							,							UpperCAmelCase_)
         a					:							List[str]							= self._trailer
         a					:							int							= self._header
         a					:							Union[str, Any]							= 0





    def __len__(					self     :						Union[str, Any]):


         """simple docstring"""







         return self._size





    def    SCREAMING_SNAKE_CASE_     (					self     :						List[Any]):


         """simple docstring"""







         return self.__len__() == 0





    def    SCREAMING_SNAKE_CASE_     (					self     :						Any							,							UpperCAmelCase_     :						int							,							UpperCAmelCase_     :						Optional[Any]							,							UpperCAmelCase_     :						str):


         """simple docstring"""







         a					:							List[Any]							= self._Node(UpperCAmelCase_							,							UpperCAmelCase_							,							UpperCAmelCase_)
         a					:							Union[str, Any]							= new_node
         a					:							List[str]							= new_node
         self._size += 1
         return self







    def    SCREAMING_SNAKE_CASE_     (					self     :						Optional[int]							,							UpperCAmelCase_     :						List[str]):


         """simple docstring"""







         a					:							Tuple							= node._prev
         a					:							Union[str, Any]							= node._next

         a					:							Optional[Any]							= successor
         a					:							Optional[int]							= predecessor
         self._size -= 1
         a					:							int							= node._data
         a					:							List[Any]							= None
         del node
         return temp






class   UpperCamelCase					(       _DoublyLinkedBase       ):







    """simple docstring"""





    def    SCREAMING_SNAKE_CASE_     (					self     :						Dict):


         """simple docstring"""







         if self.is_empty():
              raise Exception('List is empty')
         return self._header._next._data





    def    SCREAMING_SNAKE_CASE_     (					self     :						Optional[Any]):


         """simple docstring"""







         if self.is_empty():
              raise Exception('List is empty')
         return self._trailer._prev._data





    def    SCREAMING_SNAKE_CASE_     (					self     :						str							,							UpperCAmelCase_     :						List[str]):


         """simple docstring"""







         return self._insert(self._header							,							UpperCAmelCase_							,							self._header._next)





    def    SCREAMING_SNAKE_CASE_     (					self     :						int							,							UpperCAmelCase_     :						Any):


         """simple docstring"""







         return self._insert(self._trailer._prev							,							UpperCAmelCase_							,							self._trailer)





    def    SCREAMING_SNAKE_CASE_     (					self     :						int):


         """simple docstring"""







         if self.is_empty():
              raise IndexError('remove_first from empty list')
         return self._delete(self._header._next)







    def    SCREAMING_SNAKE_CASE_     (					self     :						List[Any]):


         """simple docstring"""







         if self.is_empty():
              raise IndexError('remove_first from empty list')
         return self._delete(self._trailer._prev)


'''simple docstring'''





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






def SCREAMING_SNAKE_CASE__  (				snake_case       :				float  )   ->				float:







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






def SCREAMING_SNAKE_CASE__  (				snake_case       :				int  )   ->				float:







     """simple docstring"""
     a					:							List[str]							= (circles_number + 1 - sqrt(2 * circles_number  )) / (
         2 * (circles_number**2 + 1)
     )
     a					:							int							= circles_number * intersection_y

     a					:							List[Any]							= intersection_x * intersection_y / 2
     a					:							List[str]							= circle_bottom_arc_integral(
         1 / 2  ) - circle_bottom_arc_integral(snake_case  )

     return triangle_area + concave_region_area






def SCREAMING_SNAKE_CASE__  (				snake_case       :				float = 1 / 1_000  )   ->				int:







     """simple docstring"""
     a					:							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() = }''')

