



from __future__ import annotations

from queue import Queue
def 						snake_case__							(       __lowercase				,  __lowercase							)		->		tuple[int, int]:







						"""simple docstring"""

						a ^= b
						b ^= a
						a ^= b
						return a, b
def 						snake_case__							(       __lowercase				,  __lowercase							)		->		list[list[int]]:







						"""simple docstring"""

						A__						:				Any         =			1
						while (1 << j) < max_node:
												for i in range(1				,  max_node + 1							):
																		A__						:				str         =			parent[j - 1][parent[j - 1][i]]
												j += 1
						return parent
def 						snake_case__							(       __lowercase				,  __lowercase				,  __lowercase				,  __lowercase							)		->		int:







						"""simple docstring"""

						if level[u] < level[v]:
												A__     ,							A__						:				List[Any]         =			swap(__lowercase				,  __lowercase							)
						# making depth of u same as depth of v
						for i in range(1_8				,  -1				,  -1							):
												if level[u] - (1 << i) >= level[v]:
																		A__						:				Tuple         =			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(1_8				,  -1				,  -1							):
												if parent[i][u] not in [0, parent[i][v]]:
																		A__     ,							A__						:				Dict         =			parent[i][u], parent[i][v]
    # returning longest common ancestor of u,v
						return parent[0][u]
def 						snake_case__							(       __lowercase				,  __lowercase				,  __lowercase				,  __lowercase				,  __lowercase = 1				,  )		->		tuple[list[int], list[list[int]]]:







						"""simple docstring"""

						A__						:				int         =			0
						A__						:				Queue[int]         =			Queue(maxsize=__lowercase							)
						q.put(__lowercase							)
						while q.qsize() != 0:
												A__						:				Optional[Any]         =			q.get()
												for v in graph[u]:
																		if level[v] == -1:
																								A__						:				Dict         =			level[u] + 1
																								q.put(__lowercase							)
																								A__						:				Union[str, Any]         =			u
						return level, parent
def 						snake_case__							(       )		->		None:







						"""simple docstring"""

						A__						:				Optional[int]         =			1_3
						# initializing with 0
						A__						:				Any         =			[[0 for _ in range(max_node + 1_0							)] for _ in range(2_0							)]
						# initializing with -1 which means every node is unvisited
						A__						:				str         =			[-1 for _ in range(max_node + 1_0							)]
						A__						:				dict[int, list[int]]         =			{
						    1: [2, 3, 4],
						    2: [5],
						    3: [6, 7],
						    4: [8],
						    5: [9, 1_0],
						    6: [1_1],
						    7: [],
						    8: [1_2, 1_3],
						    9: [],
						    1_0: [],
						    1_1: [],
						    1_2: [],
						    1_3: [],
						}
						A__     ,							A__						:				Optional[int]         =			breadth_first_search(__lowercase				,  __lowercase				,  __lowercase				,  __lowercase				,  1							)
						A__						:				Dict         =			create_sparse(__lowercase				,  __lowercase							)
						print("LCA of node 1 and 3 is: "				,  lowest_common_ancestor(1				,  3				,  __lowercase				,  __lowercase							)							)
						print("LCA of node 5 and 6 is: "				,  lowest_common_ancestor(5				,  6				,  __lowercase				,  __lowercase							)							)
						print("LCA of node 7 and 11 is: "				,  lowest_common_ancestor(7				,  1_1				,  __lowercase				,  __lowercase							)							)
						print("LCA of node 6 and 7 is: "				,  lowest_common_ancestor(6				,  7				,  __lowercase				,  __lowercase							)							)
						print("LCA of node 4 and 12 is: "				,  lowest_common_ancestor(4				,  1_2				,  __lowercase				,  __lowercase							)							)
						print("LCA of node 8 and 8 is: "				,  lowest_common_ancestor(8				,  8				,  __lowercase				,  __lowercase							)							)


if __name__ == "__main__":
		main()




from __future__ import annotations
def 						snake_case__							(       __lowercase				,  __lowercase				,  __lowercase				,  )		->		list[float | int]:







						"""simple docstring"""

						if not all((nth_term, start_term_a, common_ratio_r)							):
												return []
						A__						:				list[float | int]         =			[]
						A__						:				Optional[Any]         =			1
						A__						:				List[Any]         =			common_ratio_r
						for _ in range(int(__lowercase							)							):
												if not series:
																		series.append(__lowercase							)
												else:
																		power += 1
																		series.append(float(start_term_a * multiple							)							)
																		A__						:				Optional[Any]         =			pow(float(__lowercase							)				,  __lowercase							)
						return series


if __name__ == "__main__":
		import doctest

		doctest.testmod()

		snake_case     :    Optional[int]						=    float(input('Enter the last number (n term) of the Geometric Series'))
		snake_case     :    str						=    float(input('Enter the starting term (a) of the Geometric Series'))
		snake_case     :    int						=    float(
		    input('Enter the common ratio between two terms (r) of the Geometric Series')
		)
		print('Formula of Geometric Series => a + ar + ar^2 ... +ar^n')
		print(geometric_series(nth_term, start_term_a, common_ratio_r))