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)

import java.util.Scanner
import kotlin.system.exitProcess

val MOD = 998244353

fun main(args: Array<String>) {
	val scn = Scanner(System.`in`)
	val n = scn.nextInt()
	val ds = IntArray(n){scn.nextInt()}.toMutableList()
	if (ds[0]!=0) retire()
	ds.removeAt(0)
	ds.sort()
	if (ds[0]!=1) retire()
	var ans = 1L
	var l = 1
	var cntBef = 1
	var cnt = 0
	for (d in ds){
		when(d){
			l -> cnt++
			l+1 -> {
				ans = (ans*powMod(cntBef, cnt))%MOD
				cntBef = cnt
				cnt = 1
				l++
			}
			else -> retire()
		}
	}
	if (cnt!=0) ans = (ans*powMod(cntBef, cnt))%MOD
	println(ans)
}

fun powMod(b: Int, e: Int): Long{
	if (b==1) return 1
	var ans = 1L
	var b = b.toLong()
	var e = e
	while (e!=0){
		if (e%2==1) ans = (ans*b)%MOD
		b = (b*b)%MOD
		e /= 2
	}
	return ans
}

fun retire(){
	println(0)
	exitProcess(0)
}