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 (1.3.71) (judge verdict: Accepted)

fun main(args: Array<String>){

    var N = readLine()!!.toInt()
    var D = readLine()!!.split(" ").map{it.toLong()}
    var countArray = LongArray(N){0}
    var ans :Long = 1
    val MOD = 998244353

    for(i in countArray.indices){
        countArray[D[i].toInt()]++
    }

    for(i in countArray.indices){

        for(j in 0 until countArray[i]){
            if(i != 0) ans = ans * countArray[i-1] % MOD
        }
    }

    if(D[0] != 0L){
        println(0)
    } else if(countArray[0] != 1L) {
        println(0)
    } else {
            if(ans < 0) ans += MOD
            println(ans)
        }
}