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 Python (3.8.2) (judge verdict: Wrong Answer)

#!/usr/bin/env python3
import sys

MOD = 1000000007  # type: int


def solve(N: int, A: "List[int]"):
    from functools import reduce
    xyz_list = [[0]*3]
    for a in A:        
        xyz_list.append(xyz_list[-1][:])
        try:
            xyz_list[-1][xyz_list[-1].index(a)] += 1
        except ValueError:
            return -1
    T = [xyz.count(a) for xyz, a in zip(xyz_list, A)]
    return reduce(lambda a,b: a*b%MOD, T)

# Generated by 1.1.7.1 https://github.com/kyuridenamida/atcoder-tools
def main():
    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    A = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    print(solve(N, A))

def test():
    import doctest
    doctest.testmod()

if __name__ == '__main__':
    #test()
    main()