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 std::io::Read;

const MOD : u64 = 1_000_000_007;

fn main(){
  let mut buf = String::new();
  std::io::stdin().read_to_string(&mut buf).unwrap();
  let mut iter = buf.split_whitespace().map(|s| s.parse().unwrap());
  
  let n : u64 = iter.next().unwrap();
  let A : Vec<_> = iter.collect();
  
  let s0 = ([0_u64;3], 1_u64);
  let ans = A.iter().try_fold(s0, |s, &a|{
    let (mut dp, ans) = s;

    let c = dp.iter().filter(|&s| *s == a).count();
    if c == 0 {
      return None;
    }

    let i = dp.iter().position(|s| *s == a).unwrap();
    dp[i] += 1;

    let ans = ans * (c as u64) % MOD;
    
    Some((dp, ans))
  });
  
  println!("{}", ans.map_or(0, |s| s.1 ));
}