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 C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include<iostream>

using namespace std;
const int mod=998244353;
long long ksm(long long a,long long b){
	long long res=1;
	while(b){
		if(b&1){
			res*=a;
			res%=mod;
		}
		a*=a;
		a%=mod;
		b>>=1;
	}
	return res;
}

int main(){
	ios::sync_with_stdio(0);
	cin.tie(0),cout.tie(0);
	long long n,x;
	long long a[100010];
	cin>>n;
	int flag=0;
	cin>>x;
	int ans=x;
	if(x!=0) flag=1;
	for(int i=1;i<n;i++){
		cin>>x;
		a[x]++;
		if(x==0) flag=1;
		if(x>ans){
			ans=x;
		}
	}
	
	if(flag){
		cout<<"0"<<endl;
	}
	else{
	long long sum=1;
		for(int i=1;i<ans;i++){
			if(a[i]==0){
				flag=1;
				break;
			}
			sum*=ksm(a[i],a[i+1]);
			sum%=mod;
//			cout<<"SUM:"<<sum<<endl;
		}
		if(flag) cout<<"0"<<endl;
		else cout<<sum<<endl;
	}
	return 0;
}