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 Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.util.Arrays;
import java.util.Scanner;

public class Main {

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int N = sc.nextInt();
		int A[] = new int[N];
		for (int i=0;i<N;i++) {
			A[i] = sc.nextInt();
		}
		if (A[0]!=0) {
			System.out.println(0);
			return ;
		}
		Arrays.parallelSort(A);
		if (N==1) {
			System.out.println(1);
			return ;
		}
		if (A[1]==0) {
			System.out.println(0);
			return ;
		}
		long B[] = new long[N];
		for (int i=1;i<N;i++) {
			if (A[i]!=A[i-1]&&A[i]!=A[i-1]+1) {
				System.out.println(0);
				break;
			}
			B[A[i]]++;
		}
		long count =1l;
		B[0]=1;
		for (int i=1;i<N;i++) {
			for (int j=0;j<B[i];j++) {
				count = (count *B[i-1])%998244353l;
			}
		}
		System.out.println(count);
	}
}