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)

#define _USE_MATH_DEFINES
#include <iostream>
#include <string>
#include <utility>
#include <stack>
#include <vector>
#include <queue>
#include <algorithm>
#include <map>
#include <climits>
#include <set>
#include <cmath>
#include <numeric>
#include <iomanip>

using namespace std;

long long MOD = 998244353;
int BIT_SIZE = 64;
long long power(long long m, long long n){
    long long ans = 1;
    long long tmp = m;
    for(int i = 0; i < BIT_SIZE; i++){
        //cout << ans << endl;
        if(((n >> i) & 1) == 1){
            ans = ans * tmp % MOD;
        }
        tmp = (tmp * tmp) % MOD;
    }
    return ans;
}
int counts[1000001];
int main(){ 
    int N;
    cin >> N;
    int max_d = -1;
    for(int i = 0; i < N; i++){
        int d;
        cin >> d;
        max_d = max(d, max_d);
        counts[d] ++;
    }
    if(counts[0] != 1){
        cout << 0 << endl;
        return 0;
    }
    for(int i = 0; i <= max_d; i++){
        if(counts[i] == 0){
            cout << 0 << endl;
            return 0;
        }
    }
    long long ans = 1;
    for(int i = 1; i <= max_d; i++){
        ans = (ans * power(counts[i - 1], counts[i])) % MOD;
    }
    cout << ans << endl;
    return 0;
}