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 <iostream>
#include <vector>
#include <utility>
#include <map>
using namespace std;

typedef pair<int,pair<int,int> > triple;

long MOD = 1000000007;

int n;
vector<int> A;
map<triple, int> mem;
long dp(int r, int b, int g){
	int prog = r+b+g;
	if(prog == n) return 1;
	triple c = pair<int, pair<int, int> >(r, pair<int,int>(b,g));
	if(mem.find(c) != mem.end()) return mem[c];
	int at = prog;
	long sm = 0;
	if(r == A[at]) sm += dp(r+1,b,g);
	if(b == A[at]) sm += dp(r,b+1,g);
	if(g == A[at]) sm += dp(r,b,g+1);
	return mem[c] = sm % MOD;
}

int main() {
	cin >> n;
	A = vector<int>(n);
	for(int i = 0; i < n; i++) cin >> A[i];
	cout << dp(0,0,0);
	return 0;
}