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 C# (.NET Core 3.1.201) (judge verdict: Wrong Answer)

using System;
using System.Linq;
using System.Collections;
using System.Collections.Generic;
using System.Text;


class Program
{
    static void Main(string[] args)
    {
        const long mod = 998244353;

        // 入力
        long n = long.Parse(Console.ReadLine());
        long[] d = Console.ReadLine().Split().Select(long.Parse).ToArray();

        Array.Sort(d);

        var dic = new Dictionary<long, long>();
        for (int i = 0; i < d.Length; i++)
        {
            if (!dic.ContainsKey(d[i])) dic.Add(d[i], 1);
            else dic[d[i]]++;
        }

        long ans = 1;
        long count = 0;
        long parent = 0;
        dic.OrderBy(val => val.Key);

        foreach (var item in dic)
        {
            if (item.Key == 0)
            {
                if (item.Value != 1)
                {
                    ans = 0;
                    break;
                }
            }

            if (count == item.Key)
            {
                long tmp = 1;
                for (int i = 0; i < item.Value; i++)
                {
                    tmp *= parent % mod;
                }

                if (0 < tmp) ans *= tmp % mod;
                ans %= mod;

                count++;
                parent = item.Value;
            }
            else
            {
                ans = 0;
                break;
            }
        }

        // 出力
        Console.WriteLine(ans);
    }
}