Given an undirected graph G with 12 nodes (labeled 0-11) and the following edge list:
Edges: [(3,4), (2,7), (4,7), (4,8), (4,5), (6,9), (1,9), (5,11), (2,3), (4,10), (3,10), (8,10), (3,6), (5,7), (1,11), (0,2)]

Determine whether G contains a Hamiltonian path (a path that visits every node exactly once). If one exists, output the complete path. Use a backtracking search with pruning. Include verification that the path visits all 12 nodes and every consecutive pair in the path is a valid edge.
