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<algorithm>
#include<string>
#include<map>
#include<queue>
#include<cmath>

using namespace std;

int main() {
	long long n;
	cin >> n;
	vector<long long>d(n);
	vector<long long>count(n);
	long long ans = 1;
	for (int i = 0; i < n; i++) {
		cin >> d[i];
		count[i] = 0;
	}
	sort(d.begin(), d.end());
	for (long long i = 0; i < n - 1; i++) {
		if (d[i] != d[i + 1] && d[i] + 1 != d[i + 1]) {
			d[0]=1;
		}
	}
	if (d[0] != 0) {
		cout << 0 << endl;
	}
	else {
		for (long long i = 1; i < n; i++) {
			count[d[i]]++;
		}
		
		for (long long i = 1; i < n; i++) {
			if (count[i] != 0) {
				for (long long j = 1; j < count[i + 1] + 1; j++) {
					ans = (long long)((ans * count[i]) % 998244353);
				}
			}
		}
		cout << ans%998244353<< endl;
	}
}