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 C++14 (GCC 5.4.1) (judge verdict: Accepted)

#include <iostream>
#include <map>

using namespace std;
const long MOD = 1000000007;

int main(){
    int N;
    cin >> N; 

    int A[N];
    int cnt = 0;
    for(int i=0; i<N; i++){
        cin >> A[i];
        if(A[i] == 0) cnt++;
    }

    long ans = 1;
    int pre = 3;
    map<int, int> m;
    for(int i=0; i<N; i++){
        if(m.count(A[i]) == 0){
            m[A[i]] = 1;
        }else{
            m[A[i]]++;
        }
        if(A[i] != 0){
            ans = ans * m[A[i]-1] % MOD;
            m[A[i]-1]--;
        }
    }

    int s = 3;
    while(cnt != 0){
        ans = ans * s % MOD;
        cnt--;
        s--;
    }

    printf("%ld\n", ans);

    return 0;
}