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

class MainClass
{
    static void Main()
    {
        int N = int.Parse(ReadLine());
        var D = ReadLine().Split().Select(int.Parse).ToArray();
        var A = new int[N];
        int max = 0;
        for (int i = 0; i < N; i++)
        {
            A[D[i]]++;
            max = Max(max, D[i]);
        }
        if (A[0] != 1)
        {
            WriteLine(0);return;
        }
        Modular ans = 1;
        for (int i = 0; i < max; i++)
        {
            ans *= Modular.Pow(A[i], A[i + 1]);
        }
        WriteLine((long)ans);
    }
}
class Modular
{
    private const long M = 998244353;
    private long value;
    public Modular(long value) { this.value = value; }
    public static implicit operator Modular(long a)
    {
        var m = a % M;
        return new Modular((m < 0) ? m + M : m);
    }
    public static Modular operator +(Modular a, Modular b)
    {
        return a.value + b.value;
    }
    public static Modular operator -(Modular a, Modular b)
    {
        return a.value - b.value;
    }
    public static Modular operator *(Modular a, Modular b)
    {
        return a.value * b.value;
    }
    public static Modular Pow(Modular a, long n)
    {
        switch (n)
        {
            case 0:
                return 1;
            case 1:
                return a;
            default:
                var p = Pow(a, n / 2);
                return p * p * Pow(a, n % 2);
        }
    }
    public static Modular operator /(Modular a, Modular b)
    {
        return a * Pow(b, M - 2);
    }
    private static readonly List<int> facs = new List<int> { 1 };
    private static Modular Fac(int n)
    {
        for (int i = facs.Count; i <= n; ++i)
        {
            facs.Add((int)(Math.BigMul(facs.Last(), i) % M));
        }
        return facs[n];
    }
    public static Modular Ncr(int n, int r)
    {
        return (n < r) ? 0
             : (n == r) ? 1
                        : Fac(n) / (Fac(r) * Fac(n - r));
    }
    public static explicit operator int(Modular a)
    {
        return (int)a.value;
    }
}