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;
constexpr int64_t mod=998244353;

int main()
{
  int64_t n;
  cin>>n;
  vector<int64_t> dist(n);
  for(auto&& e: dist) 
  {
    cin>>e;
  }
  map<int64_t,int64_t> map;
  for(const auto& e: dist) 
  {
    map[e]++;
  }
  vector<int64_t> vec,tmp;
  bool flag=false;
  for(const auto& e:map)
  {
    //if(e.first!=vec.size()) flag=true;
    tmp.push_back(e.first);
    vec.push_back(e.second);
  }
  for (int i = 0; i < tmp.size(); ++i)
  {
    if(tmp[i]!=i) flag = true;
  }
  if(map[0]!=1 || flag)
  {
    cout<<0<<endl;
    return 0;
  }
  int64_t ans=1;
  for (int i = 2; i < vec.size(); ++i)
  {
    ans*=pow(vec[i-1],vec[i]);
    ans%=mod;
  }
  cout<<ans<<endl;
}