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 (1.0.0) (judge verdict: Wrong Answer)

fun main(args: Array<String>) {
    sumitrust2019e()
}

private const val MOD = (1000000007).toLong()

private enum class Color { R, G, B }

fun sumitrust2019e() {
    val n = readLine()!!.toInt()
    val aList = readLine()!!.split(' ').map { it.toLong() }.toLongArray()

    var answer = 3L
    val left = mutableMapOf<Color, Long>(Color.R to 1, Color.G to 0, Color.B to 0)
    aList.drop(1).forEachIndexed { index, i ->
        val candidate = left.filterValues { it == i }
        answer = answer * candidate.size % MOD

        var changed = false
        candidate.forEach { 
            val (t,u) = it
            if (!changed) left[t] = left[t]!! + 1
            changed = true
        }
    }

    println(answer)
}