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)

//============================================================================
// Name        : b.cpp
// Author      :
// Version     :
// Copyright   : Your copyright notice
// Description : Hello World in C++, Ansi-style
//============================================================================

//============================================================================
// Name        : a.cpp
// Author      :
// Version     :
// Copyright   : Your copyright notice
// Description : Hello World in C++, Ansi-style
//============================================================================
#include <iostream>
#include<vector>
#include<algorithm>
using namespace std;
typedef long long ll;

int main() {
	int n;cin>>n;
	vector<int> d(n);
	for(int i=0;i<n;i++){
		cin>>d[i];
	}
	ll def=998244353;
	if(d[0]!=0){
		cout<<0<<endl;
		return 0;
	}
	vector<int> count(n,0);
	for(int i=0;i<n;i++){
		count[d[i]]++;
	}
	if(count[0]!=1){
		cout<<0<<endl;
	}
	ll ans=1;
	int end=0;
	for(int i=1;i<n;i++){
		for(int j=0;j<count[i];j++){
			ans*=count[i-1];
			ans%=def;
		}
	}
	cout<<ans<<endl;
	return 0;
}