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: Wrong Answer)

#include<bits/stdc++.h>
using namespace std;

using LL = long long;

const LL MOD = 998244353;

LL BMPow(int x,int n,int m=0){
	LL ans=1;LL p=x;
	if(m==0){ while(n>0){if(n&1==1){ans*=p;} p*=p; n>>=1;}}
	else{ while(n>0){if(n&1==1){ans=(ans*p)%m;} p=(p*p)%m ;n>>=1;}}
	return ans;
}
 
int main(){
	LL n;
	cin>>n;

	vector<LL> d(n);
	LL max_d = 0LL;
	for(int i=0; i<n; i++){
		cin>>d[i];
		max_d = max(max_d,d[i]);
	}

	vector<LL> node_dist_cnt(n,0LL);

	for(int i=0; i<n; i++){
		node_dist_cnt[d[i]]++;
	}

	for(int i=0; i<n; i++){
		if(node_dist_cnt[i]==0){
			if(i<max_d){
				cout<<0<<endl;
				return 0;
			}
		}
	}

	if(node_dist_cnt[0]!=1){
		cout<<0<<endl;
		return 0;
	}

	LL ans=1;
	for(int i=0; i<n; i++){
		if(node_dist_cnt[i]==0) break;
		ans=(ans*BMPow(node_dist_cnt[i],node_dist_cnt[i+1],MOD))%MOD;
	}
	cout<<ans%MOD<<endl;
	return 0;
}