Circle Coloring

You are given $N$ sets of integers $A_1, A_2, \ldots, A_N$. For each valid $i$, let's denote the elements of the set $A_i$ by $\{c_{i,1}, c_{i,2}, \ldots, c_{i,|A_i|}\}$. Find the number of ways to choose a sequence $(a_1, a_2, \ldots, a_N)$ such that: - $a_i \in A_i$ for each valid $i$ - $a_i \neq a_{i+1}$ for each valid $i$ and $a_1 \neq a_N$ Since this number may be large, compute it mo