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 Java8 (OpenJDK 1.8.0) (judge verdict: Runtime Error)

import java.io.PrintWriter;
import java.util.Scanner;

public class Main {

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		PrintWriter out = new PrintWriter(System.out);
		Integer N = Integer.parseInt(sc.next());
		int[] D = new int[N];
		int[] count = new int[N];
		long result = 1;
		for(int i=0;i<N;i++) {
			D[i] = Integer.parseInt(sc.next());
			count[D[i]] ++;
		}
		if(D[0] != 0 || count[0] != 1) {
			out.println(0);
		}
		else {
			for (int i=0;i<N;i++) {
				for(int j=0;j<N;j++) {
					result *= count[i-1];
					result %= 998244353;
				}
			}
			out.println(result);
		}

		out.flush();
	}
}