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;
const long long MOD = 998244353;

long long Mod_pow(long long N, long long P) {
    if (P == 0) return 1;
    if (P % 2 == 0) {
        long long t = Mod_pow(N, P / 2);
        return t * t % MOD;
    }
    return N * Mod_pow(N, P - 1);
}

int main(){
    int N;
    cin >> N;
    vector<int> D(N);
    int zero = 0;
    for(int i = 0; i < D.size(); i++) {
        cin >> D[i];
        if (D[i] == 0) zero++;
    }

    sort(D.begin(), D.end());
    
    long long M = D[N-1];
    bool judge = true;
    vector<long long> cnt(M + 1, 0);
    for (int i = 0; i < N; i++) cnt[D[i]]++;
    for (int i = 1; i < N; i++) {
        if (abs(D[i]-D[i-1]) > 1) judge = false;
    }
    if (zero != 1) judge = false;

    long long ans = 0;
    if (judge) {
        ans = 1;
        for (int i = 1; i < cnt.size(); i++) {
            ans *= Mod_pow(cnt[i-1], cnt[i]);
            ans %= MOD;
        }
    }
    cout << ans << endl;
}