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++ (GCC 5.4.1) (judge verdict: Runtime Error)

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


int main() {
 #define int long long
  int A,B,C,N,b,x,s;
  
  cin >> N;
  vector<int>D(N);
  vector<int>a;
  
  for(int i=0;i<N;i++){
    cin >> D.at(i);
  }
  x=D.at(0);
  std::sort(D.begin(),D.end());
  
  b=1;
  for(int i=1;i<N;i++){
    if(D.at(i)==D.at(i-1)){
      b++;
    }else if(D.at(i)==(D.at(i-1)+1)){
      a.push_back(b);
      b=1;
    }else{
      x=1;
      break;
    }
  }
  a.push_back(b);
  s=a.size();
  vector<int>c(s);
  c.at(0)=1;
  c.at(1)=1;
  for(int i=2;i<(s);i++){
    c.at(i)=1;
    for(int j=0;j<a.at(i);j++){
      c.at(i)*=a.at(i-1);
    }
  }
  A=1;
  for(int i=0;i<s;i++){
    A=A*c.at(i);
  }
  if((x!=0)||(a.at(0)!=1)){
    A=0;
  }
  B=A%(998244353);
  
cout << B << endl;

}