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: Accepted)

#include<bits/stdc++.h>
using namespace std;
#define ll long long
typedef pair<ll,ll> P;
#define M 998244353
#define all(a) (a).begin(),(a).end()
#define rep(i,n) reps(i,0,n)
#define reps(i,m,n) for(int i=(m);i<(n);i++)
int main(){
  ll n,c=1LL;cin>>n;
  vector<ll> a(n);
  map<ll,ll> m;
  rep(i,n){
    cin>>a[i];
    m[a[i]]++;
    if(i==0&&a[i]!=0){
      cout<<0;return 0;
    }
    if(i!=0&&a[i]==0){
      cout<<0;return 0;
    }
  }
  sort(all(a));
  ll e=0LL,g=1LL;
  bool f=false;
  reps(i,1,n){
    if(m[i]==0){
      cout<<0;return 0;
    }
    rep(j,m[i]){
      g=g*m[i-1]%M;
      
    }
    c+=m[i];
    if(c==n)break;
  }
  cout<<g;
}