



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

def       A_   (						__a  :					float		):

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

def       A_   (						__a  :					int		):

				"""simple docstring"""
				a__					   =		(circles_number + 1 - sqrt(2 * circles_number		)) / (
				    2 * (circles_number**2 + 1)
				)
				a__					   =		circles_number * intersection_y

				a__					   =		intersection_x * intersection_y / 2
				a__					   =		circle_bottom_arc_integral(
				    1 / 2		) - circle_bottom_arc_integral(__a		)

				return triangle_area + concave_region_area

def       A_   (						__a  :					float = 1 / 1_000		):

				"""simple docstring"""
				a__					   =		(1 - pi / 4) / 4

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

				return -1


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



"""simple docstring"""

from __future__ import annotations

from queue import Queue
def 						UpperCAmelCase   (						a_,		a_     ):





		'''simple docstring'''


		a ^= b
		b ^= a
		a ^= b
		return a, b
def 						UpperCAmelCase   (						a_,		a_     ):





		'''simple docstring'''


		lowerCamelCase	:     int            =       1
		while (1 << j) < max_node:
				for i in range(1,		max_node + 1     ):
						lowerCamelCase	:     Tuple            =       parent[j - 1][parent[j - 1][i]]
				j += 1
		return parent
def 						UpperCAmelCase   (						a_,		a_,		a_,		a_     ):





		'''simple docstring'''


		if level[u] < level[v]:
				lowerCamelCase      ,       lowerCamelCase	:     List[Any]            =       swap(a_,		a_     )
		# making depth of u same as depth of v
		for i in range(18,		-1,		-1     ):
				if level[u] - (1 << i) >= level[v]:
						lowerCamelCase	:     int            =       parent[i][u]
    # at the same depth if u==v that mean lca is found
		if u == v:
				return u
		# moving both nodes upwards till lca in found
		for i in range(18,		-1,		-1     ):
				if parent[i][u] not in [0, parent[i][v]]:
						lowerCamelCase      ,       lowerCamelCase	:     str            =       parent[i][u], parent[i][v]
    # returning longest common ancestor of u,v
		return parent[0][u]
def 						UpperCAmelCase   (						a_,		a_,		a_,		a_,		a_ = 1,		):





		'''simple docstring'''


		lowerCamelCase	:     Any            =       0
		lowerCamelCase	:     Queue[int]            =       Queue(maxsize=a_     )
		q.put(a_     )
		while q.qsize() != 0:
				lowerCamelCase	:     Tuple            =       q.get()
				for v in graph[u]:
						if level[v] == -1:
								lowerCamelCase	:     Optional[Any]            =       level[u] + 1
								q.put(a_     )
								lowerCamelCase	:     str            =       u
		return level, parent
def 						UpperCAmelCase   (						):





		'''simple docstring'''


		lowerCamelCase	:     Any            =       13
		# initializing with 0
		lowerCamelCase	:     Dict            =       [[0 for _ in range(max_node + 10     )] for _ in range(20     )]
		# initializing with -1 which means every node is unvisited
		lowerCamelCase	:     List[Any]            =       [-1 for _ in range(max_node + 10     )]
		lowerCamelCase	:     dict[int, list[int]]            =       {
		    1: [2, 3, 4],
		    2: [5],
		    3: [6, 7],
		    4: [8],
		    5: [9, 10],
		    6: [11],
		    7: [],
		    8: [12, 13],
		    9: [],
		    10: [],
		    11: [],
		    12: [],
		    13: [],
		}
		lowerCamelCase      ,       lowerCamelCase	:     Dict            =       breadth_first_search(a_,		a_,		a_,		a_,		1     )
		lowerCamelCase	:     List[Any]            =       create_sparse(a_,		a_     )
		print('LCA of node 1 and 3 is: ',		lowest_common_ancestor(1,		3,		a_,		a_     )     )
		print('LCA of node 5 and 6 is: ',		lowest_common_ancestor(5,		6,		a_,		a_     )     )
		print('LCA of node 7 and 11 is: ',		lowest_common_ancestor(7,		11,		a_,		a_     )     )
		print('LCA of node 6 and 7 is: ',		lowest_common_ancestor(6,		7,		a_,		a_     )     )
		print('LCA of node 4 and 12 is: ',		lowest_common_ancestor(4,		12,		a_,		a_     )     )
		print('LCA of node 8 and 8 is: ',		lowest_common_ancestor(8,		8,		a_,		a_     )     )


if __name__ == "__main__":
					main()






