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)

import java.util.*

fun main(args: Array<String>) {
    val sc = Scanner(System.`in`)
    val n = sc.nextInt()
    val d = (0 until n).map { sc.next().toInt() }
    println(problemnikkei20192b(n, d))
}

fun problemnikkei20192b(n: Int, d: List<Int>): Long {
    if (d[0] != 0) return 0

    val mod = fun(n: Long) = n % 998244353
    val groped = d.sorted().groupBy { it }.toList()
    var ans = 1L
    for (i in 1 until groped.size) {
        ans *= mod(Math.pow(groped[i - 1].second.size.toDouble(), groped[i].second.size.toDouble()).toLong())
        ans = mod(ans)
    }
    return ans
}