Walks, Paths and Connectivity · Hard problem

Let $G$ have vertices $1,\ldots,6$ and edges $12,23,13$ (a triangle on $1,2,3$), the single bridge $34$, and $45,56,46$ (a triangle on $4,5,6$). Among the walks of length $5$ from vertex $1$ to vertex $6$, how many of them are actual paths (repeat no vertex)?