P11075 Inequality Relations (Enhanced Version).

This problem is an enhanced version of [Inequality Relations](https://loj.ac/p/575). It is recommended that you solve the original problem before challenging this enhanced version.

Given a string $s_1,s_2,\cdots ,s_n$ consisting only of the two characters ``. Let $f(s)$ be the number of permutations $p_1,p_2,\cdots ,p_{n+1}$ such that $p_i

One line containing a positive integer $n$.

One line containing an integer, representing the number of permutations satisfying the requirement modulo $998244353$.

### Sample Explanation For the string $s1=$ ``, there is exactly one permutation $(2,1)$ that satisfies the requirement, so $f(s2)=1$. Therefore, the answer is $f(s1)+f(s2)=2$. ### Constraints | Test Point ID | $n=$ | | :----------: | :----------: | | $1$ | $1$ | | $2$ | $2$ | | $3$ | $3$ | | $4$ | $5$ | | $5$ | $10$ | | $6$ | $15$ | | $7$ | $20$ | | $8$ | $30$ | | $9$ | $50$ | | $10$ | $100$ | For all testdata, it is guaranteed that $1\le n\le 100$. Translated by ChatGPT 5