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: Accepted)

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

const long long c = 998244353;

long long power(long long x, long long y) 
{ 
    long long res = 1;
  
    x = x % c; 
    while (y > 0) 
    { 
        
        if (y & 1) 
            res = (res*x) % c; 
  
        
        y = y>>1; 
        x = (x*x) % c;   
    } 
    return res; 
} 
int main(){
  long long N ;
  cin>>N;
  vector<long long> D(N);
  for(long long i = 0 ; i < N ; i++) cin>>D[i];
  if(D[0] != 0) 
  {
  	cout<<0<<endl;
  	return 0;
  }
  if(N == 1)
  {
  	cout<<1<<endl;
  	return 0;
  }
  sort(D.begin(),D.end());
  if(D[1] == D[0])
  {
  	cout<<0<<endl;
  	return 0;
  }
  vector<long long> A(N,0);
  for(long long i = 1 ; i < N ; i++)
  {
  	A[D[i]-1]++;
  }
  long long ans = 1;
  long long temp = 1;
  bool flag = false;
  for(long long i = 0 ; i < N ; i++)
  {
  	if(A[i] == 0) flag = true;
  	else if(flag == true && A[i]!=0) 
  	{
  		cout<<0<<endl;
  		return 0;
  	}
  	else 
  	{
  		long long temp_1 = power(temp,A[i]);
  		ans = ((ans*(temp_1))%c);
  		temp = A[i];
  	}
  }
  cout<<ans<<endl;
}