

# using dfs for finding eulerian path traversal







def       A_				(   _UpperCAmelCase    ,     _UpperCAmelCase    ,     _UpperCAmelCase    ,     _UpperCAmelCase=None   ):
						SCREAMING_SNAKE_CASE_:			List[Any]  		=	(path or []) + [u]
						for v in graph[u]:
												if visited_edge[u][v] is False:
																		SCREAMING_SNAKE_CASE_		,   SCREAMING_SNAKE_CASE_:			Any  		=	True, True
																		SCREAMING_SNAKE_CASE_:			List[str]  		=	dfs(_UpperCAmelCase    ,     _UpperCAmelCase    ,     _UpperCAmelCase    ,     _UpperCAmelCase   )
						return path







def       A_				(   _UpperCAmelCase    ,     _UpperCAmelCase   ):
						SCREAMING_SNAKE_CASE_:			Union[str, Any]  		=	0
						SCREAMING_SNAKE_CASE_:			List[Any]  		=	-1
						for i in range(_UpperCAmelCase   ):
												if i not in graph.keys():
																		continue
												if len(graph[i]   ) % 2 == 1:
																		odd_degree_nodes += 1
																		SCREAMING_SNAKE_CASE_:			Optional[Any]  		=	i
						if odd_degree_nodes == 0:
												return 1, odd_node
						if odd_degree_nodes == 2:
												return 2, odd_node
						return 3, odd_node







def       A_				(   _UpperCAmelCase    ,     _UpperCAmelCase   ):
						SCREAMING_SNAKE_CASE_:			Union[str, Any]  		=	[[False for _ in range(max_node + 1   )] for _ in range(max_node + 1   )]
						SCREAMING_SNAKE_CASE_		,   SCREAMING_SNAKE_CASE_:			List[Any]  		=	check_circuit_or_path(_UpperCAmelCase    ,     _UpperCAmelCase   )
						if check == 3:
												print("graph is not Eulerian"   )
												print("no path"   )
												return
						SCREAMING_SNAKE_CASE_:			Tuple  		=	1
						if check == 2:
												SCREAMING_SNAKE_CASE_:			str  		=	odd_node
												print("graph has a Euler path"   )
						if check == 1:
												print("graph has a Euler cycle"   )
						SCREAMING_SNAKE_CASE_:			Optional[int]  		=	dfs(_UpperCAmelCase    ,     _UpperCAmelCase    ,     _UpperCAmelCase   )
						print(_UpperCAmelCase   )







def       A_				(   ):
						SCREAMING_SNAKE_CASE_:			Union[str, Any]  		=	{1: [2, 3, 4], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [4]}
						SCREAMING_SNAKE_CASE_:			List[Any]  		=	{1: [2, 3, 4, 5], 2: [1, 3], 3: [1, 2], 4: [1, 5], 5: [1, 4]}
						SCREAMING_SNAKE_CASE_:			str  		=	{1: [2, 3, 4], 2: [1, 3, 4], 3: [1, 2], 4: [1, 2, 5], 5: [4]}
						SCREAMING_SNAKE_CASE_:			Any  		=	{1: [2, 3], 2: [1, 3], 3: [1, 2]}
						SCREAMING_SNAKE_CASE_:			str  		=	{
						    1: [],
						    2: []
						    # all degree is zero
						}
						SCREAMING_SNAKE_CASE_:			List[str]  		=	10
						check_euler(_UpperCAmelCase    ,     _UpperCAmelCase   )
						check_euler(_UpperCAmelCase    ,     _UpperCAmelCase   )
						check_euler(_UpperCAmelCase    ,     _UpperCAmelCase   )
						check_euler(_UpperCAmelCase    ,     _UpperCAmelCase   )
						check_euler(_UpperCAmelCase    ,     _UpperCAmelCase   )


if __name__ == "__main__":
						main()









def       A_				(   _UpperCAmelCase = 10**9   ):
						SCREAMING_SNAKE_CASE_:			List[str]  		=	1
						SCREAMING_SNAKE_CASE_:			Optional[int]  		=	2

						SCREAMING_SNAKE_CASE_:			int  		=	0
						SCREAMING_SNAKE_CASE_:			Dict  		=	0
						SCREAMING_SNAKE_CASE_:			List[str]  		=	0
						while perimeter <= max_perimeter:
												perimeters_sum += perimeter

												prev_value += 2 * value
												value += prev_value

												SCREAMING_SNAKE_CASE_:			Any  		=	2 * value + 2 if i % 2 == 0 else 2 * value - 2
												i += 1

						return perimeters_sum


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






