Score : $1400$ points

Problem Statement

You are given an integer sequence $a$ of length $N$. How many permutations $p$ of the integers $1$ through $N$ satisfy the following condition?

• For each $1 \leq i \leq N$, at least one of the following holds: $p_i = a_i$ and $p_{p_i} = a_i$.

Find the count modulo $10^9 + 7$.

Constraints

• $1 \leq N \leq 10^5$
• $a_i$ is an integer.
• $1 \leq a_i \leq N$

Input

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

$N$

$a_1$ $a_2$ $...$ $a_N$

Output

Print the number of the permutations $p$ that satisfy the condition, modulo $10^9 + 7$.

3
1 2 3

4

The following four permutations satisfy the condition:

• $(1, 2, 3)$
• $(1, 3, 2)$
• $(3, 2, 1)$
• $(2, 1, 3)$

For example, $(1, 3, 2)$ satisfies the condition because $p_1 = 1$, $p_{p_2} = 2$ and $p_{p_3} = 3$.

2
1 1

1

The following one permutation satisfies the condition:

• $(2, 1)$

3
2 1 1

2

The following two permutations satisfy the condition:

• $(2, 3, 1)$
• $(3, 1, 2)$

3
1 1 1

0

13
2 1 4 3 6 7 5 9 10 8 8 9 11

6