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Contest: Task: Related: TaskD TaskF

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 $1 ≤ i ≤ N$, 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

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

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 $10^9 + 7$.

Sample Input 1

3
1 2 3

Sample Output 1

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 $p_{p_3} = 3$.

Sample Input 2

2
1 1

Sample Output 2

1

The following one permutation satisfies the condition:

  • $(2, 1)$

Sample Input 3

3
2 1 1

Sample Output 3

2

The following two permutations satisfy the condition:

  • $(2, 3, 1)$
  • $(3, 1, 2)$

Sample Input 4

3
1 1 1

Sample Output 4

0

Sample Input 5

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

Sample Output 5

6