Score : $600$ points

Problem Statement

$N$ coins numbered $0,1,\ldots,N-1$ are arranged in a row. Initially, all coins are face up. Also, you are given a sequence $A$ of length $N$ consisting of integers between $0$ and $N-1$.

Snuke will choose a permutation $p=(p_1,p_2,\ldots,p_N)$ of $(1,\ldots,N)$ at equal probability and perform $N$ operations. In the $i$-th $(1\leq i \leq N)$ operation,

• he flips $(A_{p_i}+1)$ coins: coin $(i-1) \bmod N$, coin $(i-1+1 ) \bmod N$, $\ldots$, and coin $(i -1+ A_{p_i}) \bmod N$.

After the $N$ operations, Snuke receives $k$ yen (the currency in Japan) from his mother, where $k$ is the number of face-up coins.

Find the expected value, modulo $998244353$, of the money Snuke will receive.

Constraints

• $1 \leq N\leq 2\times 10^5$
• $0\leq A_i \leq N-1$
• All values in the input are integers.

Input

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

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the answer.

2
0 1

1

$p$ can be either $(1,2)$ or $(2,1)$.

• If $(1,2)$ is chosen as $p$:

In the first operation, coin $0$ is flipped, and in the second operation, coin $1$ and coin $0$ are flipped. One coin, coin $0$, results in being face up, so he receives $1$ yen.

• If $(2,1)$ is chosen as $p$:

In the first operation, coin $0$ and coin $1$ are flipped, and in the second operation, coin $1$ is flipped. One coin, coin $1$, results in being face up, so he receives $1$ yen.

Therefore, the expected value of the money he receives is $1$ yen.

4
3 1 1 2

665496237

Print the expected value modulo $998244353$.