Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

Solution in Kotlin (1.0.0) (judge verdict: Wrong Answer)

fun main(args: Array<String>) {
    nikkei20192qualb()
}

fun nikkei20192qualb() {
    val n = readLine()!!.toInt()
    val d = readLine()!!.split(" ").map { it.toInt() }
    if (d.first() != 0) {
        println("0")
        return
    }
    // これはKotlin1.1から
    // val appear = d.groupingBy { it }.eachCount()
    val appear = mutableMapOf<Int, Int>()
    var destinationMax = 0
    d.forEach {
        if (destinationMax < it) destinationMax = it
        if (it in appear) {
            appear[it] = appear[it]!! + 1
        } else {
            appear[it] = 1
        }
    }
    var pattern = 1L
    for (i in 1..destinationMax) {
        val p = appear[i - 1] ?: 0
        val c = appear[i] ?: 0
        if (p == 0 || c == 0) {
            println(0)
            return
        }
        var f = 1
        repeat(c) { f *= p }
        pattern *= f
    }
    println(pattern % 998244353L)
}