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;
#define rep(i,s,k) for(int i=s;i<k;i++)
#define int long long
signed main(void){
    int n,m,a,b,c;
    int ans=1;
    cin>>n;
    int num[n];
    m=0;
    rep(i,0,n){
        cin>>num[i];
        if(num[i]>m)
        m=num[i];
        if(num[0]!=0){
            cout<<"0"<<endl;
            return 0;
        }
    }
    int tree[m+1];
    rep(i,0,m+1) tree[i]=0;
    rep(i,0,n) tree[num[i]]++;
    rep(i,0,m+1){
        if(i==0&&tree[0]>1){
            cout<<"0"<<endl;
            return 0;
        }
        if(tree[i]==0){
            cout<<"0"<<endl;
            return 0;
        }
        if(i>0){
            int t=1;
            rep(j,0,tree[i]){ t=t*tree[i-1];}
            ans=(ans*t)%998244353;
        }
    }
    cout<<ans%998244353<<endl;
   
    return 0;
    
    
}