Walks, Paths and Connectivity · Hard problem

The complete bipartite graph $K_{2,3}$ has parts $\{1,2\}$ and $\{3,4,5\}$, with every vertex of one part joined to every vertex of the other (edges $13,14,15,23,24,25$). How many distinct paths (no repeated vertex) visit all $5$ vertices, counting a path and its reverse as the same path?