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: Accepted)

#include <bits/stdc++.h>
using namespace std;

int main(){
    long long waru = 998244353;
    int n;
    cin >> n;
    vector<int> a(n);
    for(int i = 0; i < n; i++){
        cin >> a[i];
    }
    if(a[0] != 0){
        cout << "0" << endl;
        return 0;
    }
    vector<long long> karisum(n,0);
    int x;
    for(int i = 1; i < n; i++){
        x = a[i];
        if(x == 0){
            cout << "0" << endl;
            return 0;
        }
        karisum[x]++;
    }
    long long ans = 1;
    karisum[0] = 1;
    bool check = false;
    for(int i = 1; i < n; i++){
        if(karisum[i] == 0){
            check = true;
            continue;
        }
        if(check){
            cout << "0" << endl;
            return 0;
        }
        int temp = karisum[i];
        while(temp != 0){
            ans = ans * karisum[i-1] % waru;
            temp-=1;
        }
    }
    cout << ans << endl;
}