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>

using namespace std;

typedef long long ll;

const ll MOD = 998244353;

int main()
{
    ios_base::sync_with_stdio(0);
    int n;
    cin >> n;
    vector<ll> il(n, 0);
    vector<ll> d(n);
    for (int i = 0; i < n; ++i)
    {
        cin >> d[i];
        ++il[d[i]];
    }
    sort(d.begin(), d.end());
    if (d[0] != 0)
    {
        cout << "0";
        return 0;
    }
    for (int i = 1; i < n; ++i)
    {
        if (!(d[i] == d[i - 1] || d[i] == d[i - 1] + 1))
        {
            cout << "0";
            return 0;
        }
    }
    ll odp = 1;
    for (int i = 2; i < n; ++i)
        if (il[i] != 0)
            for (int j = 0; j < il[i]; ++j)
                odp = (odp * il[i - 1]) % MOD;
    cout << odp;
    return 0;
}