AT_abc462_g.Completely Wrong

There is a box containing $N$ balls numbered $1$ through $N$. Each ball is painted with a color represented as an integer, and ball $i$ is painted with color $C_i$.

Takahashi performed $N$ operations. The $k$-th operation $(1\le k\le N)$ is as follows:

• Draw one ball uniformly at random from the box and discard it. If the drawn ball has color $G_k$, gain $1$ point.

Find the probability, modulo $998244353$, that the total score over $N$ operations is $0$ points.

Definition of probability modulo $998244353$

It can be proved that the sought probability is always a rational number. Furthermore, under the constraints of this problem, when the rational number is expressed as an irreducible fraction $\frac{P}{Q}$, it can be proved that $Q {{}\not\equiv{}} 0 \pmod{998244353}$. Thus, there exists a unique integer $R$ satisfying $R \times Q \equiv P \pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.

盒中有 $N$ 个编号为 $1$ 至 $N$ 的球。每个球被涂上一种颜色，颜色用整数表示；其中第 $i$ 个球的颜色为 $C_i$。

高桥进行了 $N$ 次操作。第 $k$ 次操作（$1 \le k \le N$）如下：

• 从盒中均匀随机抽取一个球并丢弃它。若抽到的球颜色为 $G_k$，则获得 $1$ 分。

求在全部 $N$ 次操作中总得分为 $0$ 分的概率（对 $998244353$ 取模）。

概率对 $998244353$ 取模的定义

可以证明：所求概率恒为有理数。此外，在本题约束下，当该有理数表示为既约分数 $\frac{P}{Q}$ 时，可证 $Q {{}\not\equiv{}} 0 \pmod{998244353}$。因此，存在唯一整数 $R$ 满足 $R \times Q \equiv P \pmod{998244353}$ 且 $0 \leq R \lt 998244353$。请输出该 $R$。

The input is given from Standard Input in the following format:

$N$
$C_1$ $C_2$ $\ldots$ $C_N$
$G_1$ $G_2$ $\ldots$ $G_N$

输入从标准输入中按以下格式给出：

$N$
$C_1$ $C_2$ $\ldots$ $C_N$
$G_1$ $G_2$ $\ldots$ $G_N$

Output the sought probability modulo $998244353$.

输出所求概率对 $998244353$ 取模的结果。

Sample 1 Explanation:
For example, the operations may proceed as follows:

• First operation: A ball with color $2$ is drawn from the box.
• Second operation: A ball with color $1$ is drawn from the box. Gain $1$ point.
• Third operation: A ball with color $3$ is drawn from the box.

In this case, the total score is $1$ point.

The probability that Takahashi's total score is $0$ is $\displaystyle \frac13$.

Sample 2 Explanation:
Takahashi's total score is always $1$ point.

Constraints

• $1\le N\le 2\times 10^5$
• $1\le C_i,G_k\le N$
• All input values are integers.

样例 1 解释：
例如，操作过程可能如下所示：

• 第一次操作：从盒子中抽出一个颜色为 $2$ 的球。
• 第二次操作：从盒子中抽出一个颜色为 $1$ 的球，获得 $1$ 分。
• 第三次操作：从盒子中抽出一个颜色为 $3$ 的球。

此时，总得分为 $1$ 分。

高桥的总得分等于 $0$ 的概率为 $\displaystyle \frac{1}{3}$。

样例 2 解释：
高桥的总得分恒为 $1$ 分。

限制条件

• $1\le N\le 2\times 10^5$
• $1\le C_i,G_k\le N$
• 所有输入值均为整数。