P5373 [Template] Polynomial Composition Function.

One day, NaCly_Fish saw $\mathsf r \color{red} \mathsf{qy}$ say in the group chat: “I finally finished writing polynomial composition! qwq”. So she curiously asked $\mathsf r \color{red} \mathsf{qy}$: “How do you write this thing?”. $\mathsf r \color{red} \mathsf{qy}$ only threw her a PDF in “Yingwen” (pinyin), but she could not understand it at all. So she asked you for help, hoping you can help her solve this difficult problem.

Given an $n$-degree polynomial $F(x)$ and an $m$-degree polynomial $G(x)$, you need to find an $n$-degree polynomial $H(x)$ that satisfies: $$H(x) \equiv F(G(x))\space (\text{mod }x^{n+1})$$ In other words, the polynomial you need should satisfy: $$H(x) \equiv \sum\limits_{i=0}^n [x^i]F(x)\times G(x)^i \space (\text{mod }x^{n+1})$$ Take all coefficients of the result modulo $998244353$.

The first line contains two positive integers $n,m$, representing the degrees of $F(x)$ and $G(x)$. The second line contains $n+1$ non-negative integers $f_i$, representing the coefficient of the $x^i$ term of $F(x)$. The third line contains $m+1$ non-negative integers $g_i$, representing the coefficient of the $x^i$ term of $G(x)$.

Output one line with $n+1$ non-negative integers, from low degree to high degree, representing the coefficients of $H(x)$.

**Constraints:** $1\le m \le n \le 20000$ $f_i,g_i \in [0,998244353)\cap \mathbb Z$ Translated by ChatGPT 5