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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;
using static System.Console;
using static System.Math;
using static MyIO;
using static MyUtil;

public class E
{
	public static void Main() => (new Solver()).Solve();
}


public class Solver
{
	public void Solve()
	{
		int N = GetInt();
		int[] A = GetInts(N);

		int[] cnt = new int[3] { 1, 0, 0 };
		var ans = new ModInt(3);
		for(int i = 1; i < N; i++)
		{
			int cand = -1;
			int candCnt = 0;
			for(int j = 0; j < 3; j++)
			{
				if(cnt[j] == A[i])
				{
					cand = j;
					candCnt++;
				}
			}
			if(cand == -1)
			{
				ans *= 0;
				break;
			}
			ans *= candCnt;
			cnt[cand]++;
		}

		WriteLine(ans.Val);
	}
}

public struct ModInt
{
	public long Mod { get; private set; }
	public long Val { get; private set; }

	public ModInt(long v)
	{
		// must initialize (default: Mod = 0)
		Mod = 1000000007;
		Val = v;
	}

	public static ModInt operator+(ModInt x, ModInt y)
	{
		long mod = x.Mod;
		long tmp = x.Val + y.Val;
		if(tmp >= mod)
			tmp -= mod;
		return new ModInt(tmp);
	}


	public static ModInt operator-(ModInt x, ModInt y)
	{
		long mod = x.Mod;
		long tmp = x.Val;
		if(tmp < y.Val)
			tmp += mod;
		tmp -= y.Val;
		return new ModInt(tmp);
	}

	public static ModInt operator*(ModInt x, ModInt y)
	{
		long mod = x.Mod;
		return new ModInt((x.Val * y.Val) % mod);
	}

	public static ModInt operator/(ModInt x, ModInt y)
	{
		long mod = x.Mod;
		long pow = mod - 2;
		long yy = y.Val;
		long inv = 1;
		while(pow > 0)
		{
			if((pow & 1) == 1)
				inv = (inv * yy) % mod;
			yy = (yy * yy) % mod;
			pow >>= 1;
		}
		return new ModInt((x.Val * inv) % mod);
	}

	public static ModInt operator+(ModInt x, long y) => x + new ModInt(y);
	public static ModInt operator-(ModInt x, long y) => x - new ModInt(y);
	public static ModInt operator*(ModInt x, long y) => x * new ModInt(y);
	public static ModInt operator/(ModInt x, long y) => x / new ModInt(y);
	public static ModInt operator++(ModInt x) => x + new ModInt(1);
	public static ModInt operator--(ModInt x) => x - new ModInt(1);
}

public static class MyIO
{
	private static string[] args = null;
	private static int num = -1;
	private static int used = -1;

	private static string getArg()
	{
		if(used == num)
		{
			args = Console.ReadLine().Split(' ');
			num = args.Length;
			used = 0;
		}
		return args[used++];
	}

	public static int GetInt() => int.Parse(getArg());
	public static long GetLong() => long.Parse(getArg());
	public static double GetDouble() => double.Parse(getArg());
	public static string GetString() => getArg();
	public static char GetChar() => getArg()[0];
	public static int[] GetInts(int N) => Enumerable.Range(0, N).Select(_ => GetInt()).ToArray();
	public static long[] GetLongs(int N) => Enumerable.Range(0, N).Select(_ => GetLong()).ToArray();
	public static double[] GetDoubles(int N) => Enumerable.Range(0, N).Select(_ => GetDouble()).ToArray();
	public static string[] GetStrings(int N) => Enumerable.Range(0, N).Select(_ => GetString()).ToArray();
}

public static class MyUtil
{
	public static void chmax<T>(ref T x, T y) where T : IComparable<T>
	{
		if(x.CompareTo(y) < 0)
			x = y;
	}

	public static void chmin<T>(ref T x, T y) where T : IComparable<T>
	{
		if(x.CompareTo(y) > 0)
			x = y;
	}

	public static void swap<T>(ref T x, ref T y)
	{
		T tmp = x;
		x = y;
		y = tmp;
	}
}