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: Runtime Error)

fun main(args: Array<String>){
  val n = readLine()!!.toInt()
  val D = readLine()!!.split(" ").map(String::toInt)
  var nodeNum = Array<Int>(100, {i -> 0})
  var ans = 1
  if(D[0] != 0){
    println(0)
    System.exit(0)
  }
  for(d in D){
  	nodeNum[d] += 1
  }
  var prev = 1
  for(node in nodeNum){
    if(prev == 0){
      break
    }
    ans *= prev.pow(node)
    prev = node
  }
  println(ans)
}
fun Int.pow(num: Int): Int{
  var ret = 1
  for(i in 1..num){
    ret *= this
  }
  return ret
}