P7439 "KrOI2021" Feux Follets (Weakened Version).

Let $\text{cyc}_\pi$ be the number of cycles when treating a permutation $\pi$ of length $n$ as a permutation mapping. Given two integers $n, k$ and a degree-$(k-1)$ polynomial, compute: $$ \sum\limits_{\pi} F(\text{cyc}_{\pi}) $$ where $\pi$ ranges over all permutations of length $n$ such that there is no position $i$ with $\pi_i = i$.

The first line contains two integers $n$ and $k$. The second line contains $k$ integers, giving the coefficients of the polynomial from low degree to high degree.

Output one integer, the answer modulo $998244353$.

### Constraints For $100\%$ of the testdata, $1 \le n, k \le 10^5$. Translated by ChatGPT 5