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.Text;
using System.Threading.Tasks;
using static System.Console;

namespace atcoder {
    class Program {
        static void Main(string[] args) {
            int N;
            N = int.Parse(ReadLine());

            var D = new List<string>(ReadLine().Split(' ')).Select(int.Parse).ToList<int>();
            long max = D.Max();

            var sorted = D.GroupBy(e => e).Select(e => new { key = e.Key, count = e.Count() }).OrderBy(e => e.key).ToList();
            if(!sorted.Any(e => e.key == 0)) {
                //WriteLine("0がない");
                WriteLine(0);
                goto ProgramEndPoint;
            }
            if(sorted[0].count != 1) {
                WriteLine(0);
                goto ProgramEndPoint;
            }
            /*
            if(sorted.Where(e => e.key != 0).Count() == 0) {
                WriteLine(0);
                goto ProgramEndPoint;
            }
            */
            int prev = -1;
            int prev_count = 1;
            Decimal ans = 1;
            foreach(var e in sorted) {
                if(e.key != prev + 1) {
                    //WriteLine("なめらかじゃない" + e.key);
                    WriteLine(0);
                    goto ProgramEndPoint;
                }

                ans *= (long)Math.Pow(prev_count,e.count);
                ans %= 998244353;
                prev = e.key;
                prev_count = e.count;
            }

            WriteLine(ans);

ProgramEndPoint:;
            ReadLine();
        }
    }
}