



import math






def      _A				(UpperCamelCase		:      int )  ->bool:



			'''simple docstring'''







			if 1 < number < 4:
						# 2 and 3 are primes
						return True
			elif number < 2 or number % 2 == 0 or number % 3 == 0:
						# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
						return False

			# All primes number are in format of 6k +/- 1
			for i in range(5      ,			int(math.sqrt(UpperCamelCase ) + 1 )      ,			6 ):
						if number % i == 0 or number % (i + 2) == 0:
									return False
			return True






def      _A				(UpperCamelCase		:      float = 0.1 )  ->int:



			'''simple docstring'''







			lowerCamelCase__ :      Optional[Any]								=		3
			lowerCamelCase__ :      Tuple								=		3

			while primes / (2 * j - 1) >= ratio:
						for i in range(j * j + j + 1      ,			(j + 2) * (j + 2)      ,			j + 1 ):
									primes += is_prime(UpperCamelCase )
						j += 2
			return j


if __name__ == "__main__":
				import doctest

				doctest.testmod()








# Author: OMKAR PATHAK, Nwachukwu Chidiebere

# Use a Python dictionary to construct the graph.
from __future__ import annotations

from pprint import pformat
from typing import Generic, TypeVar

_lowercase     							=						TypeVar('''T''')
class 				__A					( Generic[T]      ):





			def __init__(self							,    __magic_name__ = True				):
						lowerCamelCase__ :      dict[T, list[T]]								=		{}  # dictionary of lists
						lowerCamelCase__ :      List[str]								=		directed





			def        _snake_case    (self							,    __magic_name__							,    __magic_name__				):

						if not self.directed:  # For undirected graphs
									# if both source vertex and destination vertex are both present in the
									# adjacency list, add destination vertex to source vertex list of adjacent
									# vertices and add source vertex to destination vertex list of adjacent
									# vertices.
									if source_vertex in self.adj_list and destination_vertex in self.adj_list:
												self.adj_list[source_vertex].append(__magic_name__				)
												self.adj_list[destination_vertex].append(__magic_name__				)
									# if only source vertex is present in adjacency list, add destination vertex
									# to source vertex list of adjacent vertices, then create a new vertex with
									# destination vertex as key and assign a list containing the source vertex
									# as it's first adjacent vertex.
									elif source_vertex in self.adj_list:
												self.adj_list[source_vertex].append(__magic_name__				)
												lowerCamelCase__ :      int								=		[source_vertex]
									# if only destination vertex is present in adjacency list, add source vertex
									# to destination vertex list of adjacent vertices, then create a new vertex
									# with source vertex as key and assign a list containing the source vertex
									# as it's first adjacent vertex.
									elif destination_vertex in self.adj_list:
												self.adj_list[destination_vertex].append(__magic_name__				)
												lowerCamelCase__ :      Dict								=		[destination_vertex]
									# if both source vertex and destination vertex are not present in adjacency
									# list, create a new vertex with source vertex as key and assign a list
									# containing the destination vertex as it's first adjacent vertex also
									# create a new vertex with destination vertex as key and assign a list
									# containing the source vertex as it's first adjacent vertex.
									else:
												lowerCamelCase__ :      Any								=		[destination_vertex]
												lowerCamelCase__ :      Optional[Any]								=		[source_vertex]
						else:  # For directed graphs
									# if both source vertex and destination vertex are present in adjacency
									# list, add destination vertex to source vertex list of adjacent vertices.
									if source_vertex in self.adj_list and destination_vertex in self.adj_list:
												self.adj_list[source_vertex].append(__magic_name__				)
									# if only source vertex is present in adjacency list, add destination
									# vertex to source vertex list of adjacent vertices and create a new vertex
									# with destination vertex as key, which has no adjacent vertex
									elif source_vertex in self.adj_list:
												self.adj_list[source_vertex].append(__magic_name__				)
												lowerCamelCase__ :      Optional[Any]								=		[]
									# if only destination vertex is present in adjacency list, create a new
									# vertex with source vertex as key and assign a list containing destination
									# vertex as first adjacent vertex
									elif destination_vertex in self.adj_list:
												lowerCamelCase__ :      Tuple								=		[destination_vertex]
									# if both source vertex and destination vertex are not present in adjacency
									# list, create a new vertex with source vertex as key and a list containing
									# destination vertex as it's first adjacent vertex. Then create a new vertex
									# with destination vertex as key, which has no adjacent vertex
									else:
												lowerCamelCase__ :      Optional[int]								=		[destination_vertex]
												lowerCamelCase__ :      List[Any]								=		[]

						return self





			def __repr__(self				):
						return pformat(self.adj_list				)



