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>) {
  val m = 998244353L
  val N = readLine()!!.toInt()
  val D = readLine()!!.split(" ").map { it.toLong() }.sorted()
  val DMap = D.groupBy { it }.map { it.key to it.value.size }
  println(DMap)

  if (DMap[0]?.second != 1) {
    println(0)
    return
  }

  for (i in 1 until DMap.size) {
    if (DMap[i] == null) {
      println(0)
      return
    }
  }

  var sum = 1L
  for (i in 1 until DMap.size) {
    val prev = DMap[i - 1].second
    val current = DMap[i].second
    for (i in 1..current) {
      sum *= prev
      sum %= m
    }
  }
  println(sum)
}