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 Kotlin (1.3.71) (judge verdict: Accepted)

fun intList() = readLine()?.split(" ")?.map(String::toInt) ?: TODO()
fun main() {
    val n = readLine()?.toInt() ?: return
    val a = intList()
    val m = 1000000007L
    var t = 3L
    if (a.first() != 0) {
        println("0")
        return
    }
    val numbers = hashMapOf<Int, Int>()
    numbers[0] = 1
    a.drop(1).forEach {
        val b = numbers.getOrPut(it) { 0 }
        numbers[it] = b + 1
        if (it == 0) {
            t = t * (3 - b) % m
        } else {
            t = t * (numbers.getOrDefault(it - 1, 1) - b) % m
        }
    }
    println(t.toString())
}