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	<bits/stdc++.h>
using	namespace	std;

// int/long:		-2,147,483,648 - 2,147,483,648 (-2^31 <= int < 2^31)
// long/long long:	-9,223,372,036,854,775,808 - 9,223,372,036,854,775,807 (-2^63 <= long < 2^63)

#define	INF (1<<29)
//536,870,912

// lower_bound(A.begin(), A.end(), N)
// upper_bound(...

// A.erase(unique(A.begin(), A.end()), A.end())

// bit: &/and, |/or, ^/xor, ~/not

// getline(cin, String)

#define	MOD 998244353

long	P(long A, long B) {
	long S=1;
	for (long i=0; i<B; i++) {
		S *=A;
		S %=MOD;
	}
	return S;
}

int	main() {
	int N;
	cin >>N;
	vector<int> A(N, 0);
	for (int i=0; i<N; i++)
		cin >>A.at(i);
	if (0!=A.at(0)) {
		cout <<0 <<endl;
		return 0;
	}
	sort(A.begin(), A.end());
	if ((N>1)&&(1!=A.at(1))) {
		cout <<0 <<endl;
		return 0;
	}
	long C=1;
	long C0=1;
	long S=1;
	for (int i=1; i<N; i++) {
		if (A.at(i)==A.at(i-1)) {
			C++;
		} else if (A.at(i)==A.at(i-1)+1) {
			S *=P(C0, C);
			S %=MOD;
			C0=C;
			C=1;
		} else {
			cout <<0 <<endl;
			return 0;
		}
	}
	S *=P(C0, C);
	S %=MOD;
	cout <<S <<endl;
	return 0;
}