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, n) for (int i = 0; i < (int)(n); i++)
typedef pair<int, int> P;
typedef long long ll;
ll mod = 998244353;
#include <bits/stdc++.h>
using namespace std;

ll mod_pow(int a, int b){
    ll res = 1;
    rep(i,b){
        res*=a;
        res%=mod;
    }
    return res%mod;
}
int dp[100100];
int main(){
    int n;
    cin >> n;
    int mx = 0;
    rep(i,n){
        int x;
        cin >> x;
        dp[x]++;
        mx = max(mx,x);
    }
    ll ans = 1;
    for(int i = mx;i>=0;i--){
        if(dp[i]==0){
            ans = 0;
        }
    }
    for(int i = 0;i<=n;i++){
        ans*=mod_pow(dp[i],dp[i+1]);
        ans%=mod;
    }
    cout << ans%mod << endl;
}