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;

int main(){
  int N;
  cin >> N;
  int64_t mod=998244353;
  vector<int> D(N);
  for(int i=0;i<N;i++){
    cin >> D.at(i);
  }
  if(D.at(0)!=0){
    cout << 0 << endl;
    return 0;
  }
  sort(D.begin(),D.end());
  if(D.at(1)==0){
    cout << 0 << endl;
    return 0;
  }
  map<int,int64_t> m;
  for(int i=1;i<N;i++){
    if(m.count(D.at(i))){
      m.at(D.at(i))++;
    }else{
      m[D.at(i)]=1;
    }
  }
  int64_t count=1;
  int64_t ans=1;
  for(auto p:m){
    int64_t x=count*ans;
    x%=mod;
    ans=x;
    for(int i=1;i<p.second;i++){
      ans*=x;
      ans%=mod;
    }
    ans%=mod;
    count=p.second;
  }
  cout << ans << endl;
  return 0;
}