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>
#include<string>
#include<queue>
#include<algorithm>
#include<map>
#include<set>
#include<vector>
#include<math.h>

using namespace std;

#define INF 1000000007
#define LINF (1LL << 62)

typedef long long i64;
typedef pair<i64,i64> P;


i64 n, a[101010], d[101010];
i64 mod = 998244353;

i64 mod_pow(i64 x, i64 y, i64 mod){
	i64 res = 1;
	while(y > 0){
		if(y & 1) res = res * x % mod;
		x = x*x %mod;
		y >>= 1;
	}
	return res;
}

int main(){
	cin >> n;
	for(int i = 0; i < n; i++){
		cin >> a[i];
	}
	
	if(a[0] != 0){
		cout << 0 << endl;
		return 0;
	}
	sort(a, a+n);
	i64 ans = 1, now = 1, prev = 1;
	for(int i = 1; i < n; i++){
		i64 cnt = 0, j;
		for(j = i; j < n; j++){
			if(a[j] != now) break;
			cnt++;
		}
		if(cnt == 0){
			ans = 0;
			break;
		}
		ans *= mod_pow(prev, cnt, mod);
		ans %= mod;
		prev = cnt;
		now++;
		i = j-1;
	}
	cout << ans << endl;
	
	return 0;
}