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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.Arrays;
import java.util.Scanner;
public class Main {
@SuppressWarnings("resource")
public static void main(String[] args) {
	Scanner sc =new Scanner(System.in);
	long N=sc.nextLong();
	long[] D=new long[(int) N];
	D[0]=sc.nextLong();
	if(N==1&&D[0]!=0) {
		System.out.println(0);
		System.exit(0);
	}
	for(int i=1;i<N;i++) {
		D[i]=sc.nextLong();
		D[i]-=D[0];
	}
	D[0]=0;
	Arrays.sort(D);
	long preCount=1;
	long nowCount=1;
	long ans=1;
	int flag=0;
	for(int i=0;i<N-1;i++) {
		if(D[i+1]==0) {
			System.out.println(0);
			System.exit(0);
		}
		if(D[i+1]==D[i]) {
			nowCount++;
		}else if(D[i+1]==D[i]+1) {
			if(flag==1) {
				ans*=Math.pow(preCount, nowCount);
				preCount=nowCount;
				nowCount=1;
			}else if(flag==0) {
			preCount=nowCount;
			nowCount=1;
			flag=1;
			}
		}else {
			System.out.println(0);
			System.exit(0);
		}
	}
	ans*=Math.pow(preCount, nowCount);
	System.out.println(ans%=998244353);
}

}