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;
typedef long long ll;
#define inf 1000000007
//vector<vector<int>> vec(縦, vector<int>(横));
#define REP(i,N) for(int i=0;i<N;i++)
int gcd(int a, int b) {
    if(a>b){
        swap(a,b);
        }
    while(b%a!=0){
        b%=a;
        swap(a,b);
    }
    return a;
}

//cout << setprecision(123)<<ans;
//a=atoi(S.c_str());
//S=to_string(a);
int main(){
  int N,M,c;
  ll a=998244353,ans=1,ansans;
  bool ok=false;
  cin>>N;
  vector<int>v(N);
  vector<int>V(N,0);
  REP(i,N){
    cin>>v[i];
    c=v[i];
    V[c]++;
  }
  REP(i,N){
    if(V[i]==0){
      for(int j=i+1;j<N;j++){
        if(V[j]!=0)ok=true;
      }
      M=i;
      break;
    }
  }
      
      
  if(v[0]!=0||V[0]>1||ok){
    cout<<0<<endl;
    return 0;
  }
  
  for(int i=1;i<M-1;i++){
    if(V[i]>1){
      ans*=pow(V[i],V[i+1]);
  }
  }
  
  
  
  ansans=ans%a;
  cout<<ansans<<endl;
  
}