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 <iostream>
//#include <iomanip>
#include <algorithm>
//#include <numeric>
//#include <queue>
//#include <stack>
#include <math.h>
#include <vector>
//#include <string>
typedef long long LL;
using namespace std;
 
static LL MOD = 998244353;
 
int main(){
    int n;
    vector<int> d;
    int d_tmp;
    int max;
    LL ans = 1;
    int pre;
    int now;
    cin >> n;
    for(int i=0; i<n; i++){
        cin >> d_tmp;
        d.push_back(d_tmp);
    }
    sort(d.begin()+1, d.end());
    max = d[n-1];
    vector<int> n_cb(max+1);
    for(int i=0; i<=max; i++){
        n_cb[i]=0;
    }
    for(int i=0; i<n; i++){
        n_cb[d[i]]++;
    }
    if(n_cb[0]==1 && d[0]==0){
        for(int i=1; i<=max; i++){
            pre = n_cb[i-1];
            now = n_cb[i];
            if(now==0){
                ans *= 0;
            }
            ans *= (LL)powl(pre, now);
            ans %= MOD;
            //cout << pre << " " << now << " " << ans << endl;
        }
    }else{
        ans *= 0;
    }
    cout << ans << endl;
}