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: Accepted)

#include <iostream>
using namespace std;
typedef long long ll;
ll a[100003],r[100003];
ll ans=0,mx=0;
const ll md=998244353;
ll pw(ll x,ll n)
{
	if (x==0) return 0;
	if (n==0) return 1;
	ll q=pw(x,n/2);
	if (n%2==0) return q*q%md;
	else return q*q%md*x%md;
}
int main()
{
	int x;
	cin>>x;
	for (int i=1;i<=x;i++){
		cin>>a[i];
		r[a[i]]++;
		mx=max(mx,a[i]);
	}
	if (a[1]!=0||r[0]>1) ans=0;
	else
	{
		ans=1;
		for (int i=2;i<=mx;i++)
		{
			ans=ans*pw(r[i-1],r[i])%md;
		}
	}
	cout<<ans<<endl;
}