Walks, Paths and Connectivity · Challenge problem

Let $G$ be a $5$-cycle $1,2,3,4,5,1$ with an extra pendant leaf attached to every cycle vertex (so $6$ is attached to $1$, $7$ to $2$, $8$ to $3$, $9$ to $4$, $10$ to $5$). Find the product of the number of bridges and the number of cut vertices of $G$.