HMMT 2022 C10

Let $S$ be a set of size 11. A random 12 -tuple $\left(s_{1}, s_{2}, \ldots, s_{12}\right)$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi: S \rightarrow S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1} \neq \pi\left(s_{i}\right)$ for all $1 \leq i \leq 12$ (where $s_{13}=s_{1}$ ) can be written as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Compute $a$.