Problem:
Score: 500 points

Problem Statement

N people are standing in a queue, numbered 1, 2, 3, ..., N from front to back. Each person wears a hat, which is red, blue, or green.

The person numbered i says:

"In front of me, exactly A_i people are wearing hats with the same color as mine."

Assuming that all these statements are correct, find the number of possible combinations of colors of the N people's hats.

Since the count can be enormous, compute it modulo 1000000007.

Constraints

1 \leq N \leq 100000

0 \leq A_i \leq N-1

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 A_2 A_3 ... A_N

Output

Print the number of possible combinations of colors of the N people's hats, modulo 1000000007.

Sample Input 1

6
0 1 2 3 4 5

Sample Output 1

3

We have three possible combinations, as follows:

Red, Red, Red, Red, Red, Red

Blue, Blue, Blue, Blue, Blue, Blue

Green, Green, Green, Green, Green, Green

Sample Input 2

3
0 0 0

Sample Output 2

6

Sample Input 3

54
0 0 1 0 1 2 1 2 3 2 3 3 4 4 5 4 6 5 7 8 5 6 6 7 7 8 8 9 9 10 10 11 9 12 10 13 14 11 11 12 12 13 13 14 14 15 15 15 16 16 16 17 17 17

Sample Output 3

115295190

Solution in Rust (1.42.0) (judge verdict: Accepted)

use proconio::input;

fn main() {
    input! {
        n: usize,
        a: [i64; n],
    };
    let mut xyz = vec![vec![0, 0, 0]];
    for (i, &a_i) in a.iter().enumerate() {
        let mut xyz_next = xyz[i].clone();
        xyz_next.sort_by_key(|&v| -v);
        for v in xyz_next.iter_mut() {
            if *v == a_i {
                *v += 1;
                break;
            }
        }
        xyz.push(xyz_next);
    }
    let mut count = 1_usize;
    for (&a_i, xyz_i) in a.iter().zip(xyz.iter()) {
        count *= xyz_i.iter().filter(|&&v| v == a_i).count();
        count %= 1_000_000_007;
    }
    let ans = count;
    println!("{}", ans);
}