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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.Arrays;
import java.util.Scanner;

public class Main{
    public static void main(String[] args) {
        Scanner stdIn = new Scanner(System.in);
        long mod = 1000000007;

        int n = stdIn.nextInt();
        
        int x = -1;
        int y = -1;
        int z = -1;
        
        int a, candidate;
        long ans = 1;
        for(int i=0; i<n; i++){
            a = stdIn.nextInt();
            candidate = 0;

            if(x == a-1){
                candidate++;
            }
            if(y == a-1){
                candidate++;
            }
            if(z == a-1){
                candidate++;
            }

            if(x == a-1){
                x++;
            }else if(y == a-1){
                y++;
            }else if(z == a-1){
                z++;
            }

            ans = ans * candidate % mod;
        }

        int noStackedCounter = 0;
        if(x == -1){
            noStackedCounter++;
        }
        if(y == -1){
            noStackedCounter++;
        }
        if(z == -1){
            noStackedCounter++;
        }

        if(noStackedCounter == 2){
            ans = ans * 1 % mod;
        }else if(noStackedCounter == 1){
            if(x == -1){
                if(y == z){
                    ans = ans * 3 % mod;
                }else{
                    ans = ans * 6 % mod;
                }
            }else if(y == -1){
                if(x == z){
                    ans = ans * 3 % mod;
                }else{
                    ans = ans * 6 % mod;
                }
            }else{
                if(x == y){
                    ans = ans * 3 % mod;
                }else{
                    ans = ans * 6 % mod;
                }
            }
        }else{
            if(x == y && y == z){
                ans = ans * 1 % mod;
            }else if(x == y || y == z || z == x){
                ans = ans * 3 % mod;
            }else{
                ans = ans * 6 % mod;
            }
        }

        System.out.println(ans);
    }
}