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<algorithm>
#include<vector>
#include<cmath>
#include<map>
using namespace std;

typedef long long ll;

bool val(vector<ll>d){
	sort(d.begin(),d.end());
	for(ll i=1;i<d.size();++i){
		int diff = abs(d[i-1] - d[i]);
		if(diff >= 2){
			return false;
		}
	}
	return true;
}

int main(){
	ll n;
	cin >> n;
	vector<ll> d(n);
	vector<ll> vm(n,0);
	for(int i=0;i<n;++i){
		cin >> d[i];
//		cout << "-" <<endl;
		vm[d[i]]++;
	}
//	cout << "--" <<endl;

	if(d[0] != 0){
		cout << 0 <<endl;
		return 0;
	}
//	cout << "--" <<endl;
	if(val(d) == false || vm[0] != 1){
		cout << 0 <<endl;
		return 0;		
	}
//	cout << "--" <<endl;

	ll res = 1;
	for(ll i=2;i<vm.size();++i){
		if(vm[i] == 0) break;
		res *= pow(vm[i-1],vm[i]);
		res %= 998244353;
	}
	cout << res <<endl;
	

	return 0;
}