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<vector>
#include <string>
#include <cmath>
#include <algorithm>
#include <functional>
using namespace std;

int main(){
	long long N;
	long long sum = 1;
	cin >> N;

	vector<long long> D;
	D.resize(N);

	for (long long i = 0; i < N; i++){
		cin >> D[i];
	}

	if (D[0] != 0){
		cout << 0;
		return 0;

	}
	else{
		std::sort(D.begin(), D.end());

		for (long long i = 2; i <= D[N-1]; i++){
			long long before = 0;
			long long now = 0;
			int index = 0;

			for (long long j = index; j < N; j++){
				if (i - 1 == D[j]){
					before++;
				}

				if (i == D[j]){
					now++;
				}
				index ++;
			}

			if (before * now == 0){
				cout << 0;
				return 0;
			}
			sum *= pow(before, now);
		}
	}
	cout << sum;
	return 0;
}