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.*;

public class Main{
  public static void main(String[]args){
    Scanner sc = new Scanner(System.in);
    int N = Integer.parseInt(sc.nextLine());
    long ans = 1;
    long p = 998244353;
    List<Integer> list = new ArrayList<>();
    for(int i = 0; i < N; i++){
      int n = Integer.parseInt(sc.next());
      if((i == 0 && n != 0) || (i != 0 && n == 0)){
        System.out.println(0);
        return;
      }
      list.add(n);
    }
    Collections.sort(list);
    long count = 1;
    long count2 = 0;
    long now = 1;
    for(int i = 1; i < N; i++){
      int n = list.get(i);
      if(n == now){
        count2++;
        if(i == N-1){
          ans *= calc(count, count2);
          ans %= p;
        }
      }else{
        ans *= calc(count, count2);
        ans %= p;
        count = count2;
        count2 = 1;
        now = n;
        
        if(i == N-1){
          ans *= n;
          ans %= p;
        }
      }
    }
    System.out.println(ans);
  }
  static long calc(long a, long b){
    return (long)Math.pow(a,b);
  }
}