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# (Mono 4.6.2.0) (judge verdict: Runtime Error)

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

namespace AtCoder.Abc
{
    class QuestionB
    {
        public static void Main(string[] args)
        {
            // 整数の入力
            int n = int.Parse(Console.ReadLine());

            // 整数配列の入力
            var dArray = Console.ReadLine().Split(' ').Select(i => int.Parse(i));

            if(dArray.ElementAt(0) != 0)
            {
                Console.WriteLine(0);
                return;
            }

            Dictionary<int, int> dic = new Dictionary<int, int>();

            int max = -1;
            foreach(var d in dArray)
            {
                if(dic.ContainsKey(d))
                {
                    dic[d]++;
                }
                else
                {
                    dic.Add(d, 1);
                }
                if(max < d)
                {
                    max = d;
                }
            }
            
            if(dic[0] != 1)
            {
                Console.WriteLine(0);
                return;
            }

            int lastCount = 1;
            long result = 1;

            for(int i = 1; i <= max; i++)
            {
                if(dic[i] == 0 && i != max)
                {
                    Console.WriteLine(0);
                    return;
                }

                result *= (long)Math.Pow(lastCount, dic[i]);
                if(result > 998244353)
                {
                    result %= 998244353;
                }
                lastCount = dic[i];

            }


            Console.WriteLine(result %= 998244353);
        }
    }
}