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: Wrong Answer)

#include <vector>
#include <stack>
#include <queue>
#include <list>
#include <bitset>
#include <set>
#include <map>
#include <unordered_set>
#include <unordered_map>
#include <algorithm>
#include <numeric>
#include <iostream>
#include <iomanip>
#include <string>
#include <chrono>
#include <random>
#include <cmath>
#include <cassert>
#include <climits>
#include <cstring>
#include <cstdlib>
#include <functional>
#include <sstream>

using namespace std;



int main(int argc, char** argv) {
    ios::sync_with_stdio(false);
    cin.tie(0);


    const long long MOD = 1000000007;
    int n;
    cin >> n;
    vector<int> A(n);
    for (int i = 0; i < n; ++i) {
        cin >> A[i];
        if (A[i] > i) {
            cout << 0 << '\n';
            return 0;
        }
    }

    vector<vector<int>> pos(n + 1);
    pos[0] = {-1, -2, -3};

    for (int i = 0; i < n; ++i) {
        pos[A[i] + 1].push_back(i);
    }
    for (int i = 0; i <= n; ++i) {
        if (pos[i].size() > 3) {
            cout << 0 << '\n';
            return 0;
        }
        if (i - 1 >= 0 && pos[i].size() > pos[i - 1].size()) {
            cout << 0 << '\n';
            return 0;
        }
    }

    long long res = 1;

    for (int i = 1; i <= n; ++i) {
        if (pos[i].size() == 0) {
            break;
        }

        auto ord = pos[i];

        sort(ord.begin(), ord.end());
        long long cnt = 0;
        do {
            bool ok = true;
            for (int j = 0; j < ord.size(); ++j) {
                if (ord[j] <= pos[i - 1][j]) {
                    ok = false;
                    break;
                }
            }
            if (ok) {
                ++cnt;
            }
        } while (next_permutation(ord.begin(), ord.end()));

        (res *= cnt) %= MOD;
    }


    cout << res << '\n';


    return 0;
}