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.util.Scanner;

public class Main {

    public static void main(String[] args) {
        Scanner sc = new Scanner(System.in);
        int n = sc.nextInt();
        int[] d = new int[n];
        int[] count = new int[100000];
        int max = 0;
        for (int i = 0; i < n; i++) {
            d[i] = sc.nextInt();
            max = Math.max(max, d[i]);
        }

        for (int i = 0; i < n; i++) {
            count[d[i]]++;
        }
        if (count[0] != 1) {
            System.out.println(0);
            return;
        }
        long sum = 1;
        for (int i = 1; i <= max; i++) {
            if (count[i - 1] == 0 && i != max - 1) {
                System.out.println(0);
                return;
            }
            sum *= Math.pow(count[i], count[i + 1]) % 998244353;
        }
        System.out.println(sum % 998244353);
    }

}