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 (Clang 3.8.0) (judge verdict: Wrong Answer)

#include<iostream>
#include<cstdio>
#include<algorithm>
#include<cmath>
#include<vector>
#include<map>
#include<set>
#include<string>
#include<queue>
#include<stack>
using namespace std;
#define MOD 998244353
#define INF (1<<29)
#define LINF (1LL<<60)
#define EPS (1e-10)
typedef long long Int;
typedef pair<Int, Int> P;

Int mod_pow(Int x, Int a, Int m = MOD){
    if(a == 0)return 1;
    Int res = mod_pow(x, a / 2, m);
    res = res * res % m;
    if(a % 2)res *= x;
    return res % m;
}

Int inv(Int x, Int m = MOD){
    return mod_pow(x, m-2, m);
}
Int n, d;
Int cnt[110000];


int main(){
    cin >> n;
    Int maxd = 0;
    for(int i = 0;i < n;i++){
        cin >> d;
        cnt[d]++;
        maxd = max(maxd, d);
    }
    if(cnt[0] != 1){
        cout << 0 << endl;
        return 0;
    }
    Int ans = 1;
    for(int i = 1;i <= maxd;i++){
        ans = ans * mod_pow(cnt[i-1], cnt[i]) % MOD;
    }
    cout << ans << endl;
    return 0;
}