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# (Mono 4.6.2.0) (judge verdict: Runtime Error)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;

public class E
{
    static void Main()
    {
        string line = Console.ReadLine().Trim();
        var N = int.Parse(line);

        string[] countRecord = Console.ReadLine().Trim().Split(' ');
        var answerList = new Dictionary<int?, int?>();
        for (int i = 0; i < N; i++)
        {
            answerList.Add(i, int.Parse(countRecord[i]));
        }
        int hasuu = 1000000007;

        //計算処理
        var resultList = CalcCombination(N, answerList, hasuu);

        Console.WriteLine(resultList % hasuu);
    }

    /// <summary>
    /// 組み合わせチェック
    /// </summary>
    /// <returns></returns>
    private static int? CalcCombination(int N, Dictionary<int?, int?> answerList, int hasuu)
    {
        int? result = 1;
        var resultList = new Dictionary<int?, int?>();

        for (int i = 0; i < N; i++)
        {
            if (resultList.ContainsKey(answerList[i]))
            {
                if (answerList[i] != 0) result = result * (resultList[answerList[i] - 1] - resultList[answerList[i]]);
                resultList[answerList[i]] += 1;
            }
            else
            {
                resultList.Add(answerList[i], 1);
                if (answerList[i] != 0) result = result * resultList[answerList[i] - 1];
            }
            //Console.WriteLine(result.ToString());

            if (result > hasuu) result = result % hasuu;
        }
        switch (resultList[0])
        {
            case 1:
                result = result * 3;
                break;
            case 2:
            case 3:
                result = result * 6;
                break;
        }

        return result;
    }
}