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: Wrong Answer)

using System;
using System.Collections.Generic;
using System.Linq;
using System.IO;
using System.Text;
using static System.Console;
using static System.Math;

namespace CP
{
    class Atria
    {
        static void Main(string[] args)
        {
            int n = int.Parse(ReadLine());
            int[] s = ReadLine().Split(' ').Select(int.Parse).ToArray();
            if (s[0] != 0)
            {
                WriteLine(0);
                return;
            }
            long then = 1;
            long ans = 1;
            bool divided = false;
            Array.Sort(s);
            long nex = 0;
            long now = 0;
            for (int i = 0; i < n; i++)
            {
                if (s[i] == now)
                {
                    nex++;
                }
                else
                {
                    //
                    //処理は初めての枝分かれ後
                    if (divided)
                    {
                        /*階乗案
                        long tmp = 1;
                        for (int j = 2; j < next; j++)
                        {
                            tmp *= j;
                            tmp %= 998244353;
                        }
                        */
                        ans *= (Factorial(nex) % 998244353) * (long)Pow(2, then);
                        ans %= 998244353;
                    }
                    then = nex;
                    if (divided == false && nex >= 2)
                    {
                        divided = true;
                    }
                    //
                    now = s[i];
                    nex = 1;
                }
            }
            if(ans == 1)
            {
                ans = then;
            }
            WriteLine(ans);
        }

        /// <summary>
        /// 階乗を求める再帰関数
        /// </summary>
        /// <param name="x">x!を返す</param>
        /// <returns></returns>
        static long Factorial(long x)
        {
            if (x == 1) return 1;
            return x * Factorial(x - 1);
        }
    }
}