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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <iostream>
#include <vector>
#include <algorithm>
#include <cmath>
using namespace std;

int main()
{
    unsigned long long int n;
    cin >> n;
    vector<unsigned long long int> d(n);
    for (unsigned long long int i = 0; i < n; i++)
    {
        cin >> d[i];
    }
    if (d[0] != 0)
    {
        cout << 0;
        return 0;
    }
    sort(d.begin(), d.end());
    if (d[1] != 1)
    {
        cout << 0;
        return 0;
    }
    unsigned long long int m = *max_element(d.begin(), d.end());
    vector<unsigned long long int> num(m);
    unsigned long long int key = 1;
    for (unsigned long long int i = 1; i < n; i++)
    {
        if (d[i] == key)
        {
            num[key - 1]++;
        }
        else if (d[i] == key + 1)
        {
            key++;
            num[key - 1]++;
        }
        else
        {
            cout << 0;
            return 0;
        }
    }
    unsigned long long int ans = 1;
    for (unsigned long long int i = 1; i < m; i++)
    {
        for (unsigned long long int j = 0; j < num[i]; j++)
        {
            ans *= (num[i - 1] % 998244353);
            ans % 998244353;
            //  ans *= num[i - 1];
        }
    }

    cout << ans % 998244353;
}