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: Runtime Error)

import java.util.HashMap;
import java.util.Map;
import java.util.Scanner;

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

    long M = 1_000_000_007;
    int N = scanner.nextInt();
    int[] A = new int[N];
    for (int i = 0; i < N; i++) A[i] = scanner.nextInt();
    Map<Integer, Integer> freq = new HashMap<>();
    long res = 1;
    int countZeros = 0;
    for (int i = 0; i < N; i++) {
      if (A[i] != 0) {
        int p = freq.get(A[i] - 1);
        res *= p;
        res %= M;
        freq.put(A[i] - 1, p - 1);
      } else {
        countZeros++;
      }
      freq.put(A[i], freq.getOrDefault(A[i], 0) + 1);
    }
    if (countZeros > 1) {
      System.out.println(res * 6 % M);
    } else {
      System.out.println(res * 3 % M);
    }
  }
}