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

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.
