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

namespace Algorithm
{
    class Program
    {
        const int Mod = 998244353;
        static void Main(string[] args)
        {
            var n = int.Parse(Console.ReadLine());
            var d = Console.ReadLine().Split().Select(int.Parse).ToArray();

            var dict = new Dictionary<int, int>();
            for (var i = 0; i < n; i++)
            {
                if (dict.ContainsKey(d[i]))
                    dict[d[i]]++;
                else
                    dict[d[i]] = 1;
            }

            var total = 1L;
            var before = 1;
            var count = 0;

            foreach (var data in dict.OrderBy(z => z.Key))
            {
                if (data.Key != count)
                {
                    Console.WriteLine(0);
                    return;
                }

                if (data.Key != 0 && data.Key != 1)
                    total *= Multiple(before, data.Value);

                before = data.Value;
                count++;
            }

            Console.WriteLine(total);
        }

        static long Multiple(long val, int multiple)
        {
            var ans = 1L;
            foreach (var i in Enumerable.Range(1, multiple))
            {
                ans *= val % Mod;
            }

            return ans;
        }
    }
}