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

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.
