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 <bits/stdc++.h>
using namespace std;
typedef long long ll;
#define rep(i, a, b) for (int i = a; i < (int)b; i++)
#define all(v) v.begin(), v.end()
const int MOD = 998244353;
const int INF = 1e9;
//'A' = 65, 'Z' = 90, 'a' = 97, 'z' = 122

int main() {
    int n;
    cin >> n;
    vector<int> d(n);
    rep(i, 0, n) cin >> d[i];
    sort(all(d));
    d.push_back(1000000);
    if (d[0] >= 1 || d[1] == 0) {
        cout << 0 << endl;
        return 0;
    }
    ll ans = 1;
    int cnt = 1;
    vector<int> v;
    rep(i, 1, n) {
        if (d[i] - d[i - 1] >= 2) {
            cout << 0 << endl;
            return 0;
        }
    }
    rep(i, 2, n + 1) {
        if (d[i - 1] != d[i]) {
            v.push_back(cnt);
            cnt = 1;
        } else cnt++;
    }
    rep(i, 0, v.size() - 1) {
        rep(j, 0, v[i + 1]) {
            ans *= v[i];
            ans %= MOD;
        }
    }
    //rep(i, 0, v.size()) cout << v[i] << endl;
    cout << ans << endl;
    return 0;
}