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 <bits/stdc++.h>
using namespace std;
#define ll long long
#define all(V) V.begin(),V.end()
const ll MOD = 1000000007;

vector<int> col[100010];

int main() {
	int N, A, am = 0;
	ll ans = 1;
	cin >> N;
	for (int i = 0;i < N;i++) {
		cin >> A;
		am = max(A, am);
		col[A].push_back(i);
	}
	int osz = col[0].size(), sz, cnt = 0;
	if (osz == 1) {
		ans = 3;
	}
	else {
		ans = 6;
	}
	if (osz > 3) {
		cout << 0 << endl;
		return 0;
	}
	for (int i = 1;i <= am;i++) {
		cnt = 0;
		sz = col[i].size();
		if (sz > osz) {
			cout << 0 << endl;
			return 0;
		}
		if (sz == 1) {
			for (int j = 0;j < osz;j++) {
				if (col[i][0] > col[i - 1][j])
					cnt++;
			}
		}
		else if (sz == 2) {
			for (int j = 0;j < osz;j++) {
				for (int k = 0;k < osz;k++) {
					if (j == k)continue;
					if (col[i][0] > col[i - 1][j] && col[i][1] > col[i - 1][k]) {
						cnt++;
					}
				}
			}
		}
		else if (sz == 3) {
			for (int j = 0;j < osz;j++) {
				for (int k = 0;k < osz;k++) {
					for (int l = 0;l < osz;l++) {
						if (j == k)continue;
						if (j == l)continue;
						if (k == l)continue;
						if (col[i][0] > col[i - 1][j] && col[i][1] > col[i - 1][k] && col[i][2] > col[i - 1][l]) {
							cnt++;
						}
					}
				}
			}
		}
		else {
			cout << 0 << endl;
			return 0;
		}
		ans *= cnt;
		ans %= MOD;
		osz = sz;
	}
	cout << ans << endl;
}