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;
use std::str::FromStr;
use std::collections::HashMap;

fn input<T:FromStr>() -> T {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    line.trim().parse().ok().unwrap()
}

fn input_vec() -> Vec<u64> {
    let mut line: String = String::new();
    io::stdin().read_line(&mut line);
    line.split_whitespace().map(|s| s.parse().unwrap()).collect()
}

fn recur(i: usize, a_vec: &Vec<u64>, hats: &mut Vec<u64>, cache: &mut HashMap<(usize, Vec<u64>), u64>) -> u64 {
    // println!("{:?}", hats);
    if cache.contains_key(&(i, hats.to_vec())) {
        return *cache.get(&(i, hats.to_vec())).unwrap();
    }
    if i == a_vec.len() {
        return 1u64;
    }
    let mut res = 0u64;
    for j in 0..hats.len() {
        if hats[j] == a_vec[i] {
            let n = hats[j];
            hats[j] = n + 1;
            res += recur(i + 1, a_vec, hats, cache);
            hats[j] = n;
        }
    }
    if res >= 1000000007 {
        res %= 1000000007;
    }
    cache.insert((i, hats.to_vec()), res);
    res
}

fn main() {
    let n: u64 = input();
    let a_vec = input_vec();
    let mut cache: HashMap<(usize, Vec<u64>), u64> = HashMap::new();
    println!("{}", recur(0, &a_vec, &mut vec![0u64; 3], &mut cache));

}