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 (1.15.1) (judge verdict: Accepted)

use std::io::Read;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    let _n: usize = it.next().unwrap().parse().unwrap();
    let mut now = std::collections::BTreeMap::new();
    const MOD: u64 = 1_000_000_007;
    now.insert((0, 0, 0), 1u64);
    for s in it {
        let a: usize = s.parse().unwrap();
        let mut next = std::collections::BTreeMap::new();
        for (k, v) in now.iter() {
            let v = *v % MOD;
            if k.0 == a {
                let mut x = k.clone();
                x.0 += 1;
                *next.entry(x).or_insert(0) += v;
            }
            if k.1 == a {
                let mut x = k.clone();
                x.1 += 1;
                *next.entry(x).or_insert(0) += v;
            }
            if k.2 == a {
                let mut x = k.clone();
                x.2 += 1;
                *next.entry(x).or_insert(0) += v;
            }
        }
        now = next;
    }
    let mut ans = 0;
    for v in now.values() {
        ans += *v;
    }
    ans %= MOD;
    println!("{}", ans);
}

fn main() {
    run();
}