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 <iostream>
#include <vector>
#include <algorithm>
using namespace std;
#define MOD 998244353
long long binaryPowerMethod(long long z,long long n, long long mod){
  long long ans = 1;
  while(n != 0){
    if(n%2){
      ans*=z;
      if(mod)ans%=mod;
    }
    n/=2;
    z*=z;
    if(mod)z%=mod;
  }
  return ans;
}
int len[100000];
int main(){
  long long n;cin>>n;
  vector<long long> a(n);
  long long mx = 0;
  for(int i = 0; n > i; i++){
    cin>>a[i];
    mx = max(a[i],mx);
  }
  sort(a.begin(),a.end());
  for(int i = 0; n > i; i++){
    len[a[i]]++;
  }
  for(int i = 0; mx >= i; i++){
    if(!len[i]){
      cout << 0 << endl;
      return 0;
    }
  }
  if(len[0] != 1){
    cout << 0 << endl;
    return 0;
  }
  long long ans = 1;
  for(int i = 1; mx >= i; i++){
    ans = (ans * binaryPowerMethod(len[i-1],len[i],MOD))%MOD;
  }
  cout << ans << endl;
}