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)

#[test]
fn test_solve() {
    assert_eq!(solve("6\n0 1 2 3 4 5\n"), "3\n");
    assert_eq!(solve("3\n0 0 0\n"), "6\n");
    assert_eq!(solve("54\n0 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\n"), "115295190\n");
}

fn solve(src: &str) -> String {
    let mut sc = Scanner::from_str(src);

    let n: usize = sc.read();
    let a: Vec<usize> = (0..n).map(|_| sc.read()).collect();

    let mut x = 0;
    let mut y = 0;
    let mut z = 0;
    let mut ans = Finite(1);
    for &a in &a {
        ans *= [x, y, z].iter().filter(|&&x| x == a).count() as u64;

        if a == x {
            x += 1;
        } else if a == y {
            y += 1;
        } else if a == z {
            z += 1;
        } else {
            return "0\n".to_string();
        }
    }

    format!("{}\n", ans)
}

const MOD: u64 = 1_000_000_007;
#[derive(Copy, Clone, PartialEq, Eq, Debug)]
struct Finite(u64);
impl std::fmt::Display for Finite {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
impl std::ops::MulAssign<u64> for Finite {
    fn mul_assign(&mut self, rhs: u64) {
        self.0 *= rhs;
        self.0 %= MOD;
    }
}

fn main() {
    let mut buf = String::new();
    use std::io;
    use std::io::{Read, Write};
    io::stdin().read_to_string(&mut buf).unwrap();
    io::stdout().write_all(solve(&buf).as_bytes()).unwrap();
}

struct Scanner<'a>(std::str::SplitWhitespace<'a>);
impl<'a> Scanner<'a> {
    fn from_str(src: &str) -> Scanner {
        Scanner(src.split_whitespace())
    }
    fn get(&mut self) -> &str {
        self.0.next().unwrap()
    }
    fn read<T: std::str::FromStr>(&mut self) -> T {
        self.get().parse().ok().unwrap()
    }
}