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: Wrong Answer)

import java.util.Map;
import java.util.Scanner;
import java.util.TreeMap;

public class Main {
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		Long n = Long.valueOf(sc.nextLine());

		Map<Long, Long> map = new TreeMap<>();

		for (int i = 0; i < n; i++) {
			Long input = sc.nextLong();
			Long c = map.getOrDefault(input, 0L);
			c++;
			map.put(input, c);
		}
		sc.close();
		if (!map.containsKey(0L)) {
			System.out.println(0);
			return;
		}
		Long before = -1L;
		for (Long key : map.keySet()) {
			if (key != before + 1L) {
				System.out.println(0L);
				return;
			}
			before = key;
		}
		Long size = (long) map.values().size();
		long ans = 1L;
		for (Long i = 0L; i < size - 1L; i++) {
			Long a = map.get(i);
			Long b = map.get(i + 1l);
			ans = ans * (long) Math.pow(a, b);
		}
		System.out.println(ans % 998244353L);
	}
}