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;
#ifdef LOCAL_DEBUG
  #include "LOCAL_DEBUG.hpp"
#endif
#define int long long

signed main(){

  int n; cin >> n;
  vector<int> d(n);
  int Max = -1;
  for(int i = 0; i < n; i++){
    cin >> d[i];
    Max = max(d[i], Max);
  }

  vector<int> d2(Max+1, 0);
  for(int i = 0; i < n; i++){
    d2[d[i]]++;
  }
  if(d2[0] != 1 && d[0] != 0){
    cout << 0 << endl;
    return 0;
  }

  int ans = 1;
  for(int i = 1; i <= Max; i++){
    for(int j = 0; j < d2[i]; j++){
      ans *= d2[i-1];
      ans %= 998244353;
    }
  }
  cout << ans << endl;

  return 0;
}