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: Accepted)

fun main(args: Array<String>) {
  val m = 998244353L
  val N = readLine()!!.toInt()
  val D = readLine()!!.split(" ").map { it.toInt() }
  val maxD = D.max() ?: 0
  val DMap = D.groupBy { it }

  if (D[0] != 0) {
    println(0)
    return
  }

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

  var result = 1L
  for (i in 1..maxD) {
    val prev = DMap[i - 1]?.size ?: 0
    val current = DMap[i]?.size ?: 0
    // result *= prev^current
    for (i in 0 until current) {
      result = (result * prev) % m
    }
  }
  println(result)
}