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;
long long powmod(long long px, long long py, long long pmod){
	if(py == 0) return 1;
	if(py % 2 == 0) return powmod((px * px) % pmod, py / 2, pmod);
	else return (px * powmod(px, py - 1, pmod)) % pmod;
} 
int main() {
	// your code goes here
    long long n,i,d[100005],e,s=1,ans=1,mo=1;
    long long mod = (long long)998244353;
    cin >> n;
    for(i=0;i<n;i++){
        cin >> d[i];
        if(i==0&&d[i]!=0){
            cout << "0" << endl;
            return 0;
        }
        if(i!=0&&d[i]==0){
            cout << "0" << endl;
            return 0;
        }
    }
    sort(d,d+n);
    d[n]=1000000;
    for(i=1;i<n+1;i++){
        if(d[i-1]!=d[i]){
            if(i!=n&&d[i-1]+1!=d[i]){
                cout << "0" << endl;
                return 0; 
            }
            if(i==1) s=1;
            else {
                ans=ans*powmod(mo,s,mod)%mod;
                mo=s;
                s=1;
            }
        } else s++;
    }
    cout << ans << endl;
	return 0;
}