

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







def  UpperCAmelCase_   (_lowerCAmelCase :		float		):
			return (
			    (1 - 2 * point) * sqrt(point - point**2		) + 2 * point + asin(sqrt(1 - point		)		)
			) / 4







def  UpperCAmelCase_   (_lowerCAmelCase :		int		):
			__UpperCamelCase    :    int								=      (circles_number + 1 - sqrt(2 * circles_number		)) / (
			    2 * (circles_number**2 + 1)
			)
			__UpperCamelCase    :    Union[str, Any]								=      circles_number * intersection_y

			__UpperCamelCase    :    List[str]								=      intersection_x * intersection_y / 2
			__UpperCamelCase    :    Dict								=      circle_bottom_arc_integral(
			    1 / 2		) - circle_bottom_arc_integral(_lowerCAmelCase		)

			return triangle_area + concave_region_area







def  UpperCAmelCase_   (_lowerCAmelCase :		float = 1 / 10_00		):
			__UpperCamelCase    :    Tuple								=      (1 - pi / 4) / 4

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

			return -1


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


from __future__ import annotations







def  UpperCAmelCase_   (_lowerCAmelCase :		int     ,			_lowerCAmelCase :		list[int]     ,			_lowerCAmelCase :		list[int]     ,			_lowerCAmelCase :		int		):

			# Base Case
			if counter == 0 or capacity == 0:
						return 0

			# If weight of the nth item is more than Knapsack of capacity,
			#   then this item cannot be included in the optimal solution,
			# else return the maximum of two cases:
			#   (1) nth item included
			#   (2) not included
			if weights[counter - 1] > capacity:
						return knapsack(_lowerCAmelCase     ,			_lowerCAmelCase     ,			_lowerCAmelCase     ,			counter - 1		)
			else:
						__UpperCamelCase    :    Tuple								=      capacity - weights[counter - 1]
						__UpperCamelCase    :    List[str]								=      values[counter - 1] + knapsack(
						    _lowerCAmelCase     ,			_lowerCAmelCase     ,			_lowerCAmelCase     ,			counter - 1		)
						__UpperCamelCase    :    Dict								=      knapsack(_lowerCAmelCase     ,			_lowerCAmelCase     ,			_lowerCAmelCase     ,			counter - 1		)
						return max(_lowerCAmelCase     ,			_lowerCAmelCase		)


if __name__ == "__main__":
					import doctest

					doctest.testmod()