Description

Given polynomials $F(x)$, $A(x)$, and $B(x)$ satisfying:

$$\frac{\text dF(x)}{\text dx} \equiv A(x)\text e^{F(x)-1}+B(x) \pmod{x^n}$$

and $F(0)=1$.

Given $A(x)$ and $B(x)$, find the coefficients of the first $n$ terms of $F(x)$.

Output the answer modulo $998244353$.

Input Format

The first line contains a positive integer $n$, representing the degree of $A(x)$ and $B(x)$.
The second line contains $n+1$ integers, from low degree to high degree, representing the coefficients of $A(x)$.
The third line contains $n+1$ integers, from low degree to high degree, representing the coefficients of $B(x)$.

Output Format

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

9
2 9 8 7 3 6 5 4 1 12
23 9 8 7 4 6 1 3 2 5 

1 25 34 332748429 124783260 22560 624092696 904826719 284383572 50973515

Hint

Constraints

For $30\%$ of the testdata, $1 \le n \le 5000$.
For $100\%$ of the testdata, $1 \le n \le 10^5$.

All inputs are guaranteed to be in the range $[0,998244353)$.

Translated by ChatGPT 5