Graph Coloring · Superchallenge problem

Let $G$ be an $11$-vertex, triangle-free graph with vertices $\{v_1,v_2,v_3,v_4,v_5,u_1,u_2,u_3,u_4,u_5,w\}$ and edges: the $5$-cycle $v_1\text{-}v_2\text{-}v_3\text{-}v_4\text{-}v_5\text{-}v_1$; the shadow edges connecting each $u_i$ to $v_{i-1}$ and $v_{i+1}$ (indices $\bmod 5$, so $u_1$ connects to $v_5$ and $v_2$, etc.); and the vertex $w$ adjacent to all five $u_i$ ($20$ edges total, $\chi(G)=4$). With $4$ colors available, how many proper colorings does $G$ have?