Problem:
Score : 700 points

Problem Statement

Aoki loves numerical sequences and trees.

One day, Takahashi gave him an integer sequence of length N, a_1, a_2, ..., a_N, which made him want to construct a tree.

Aoki wants to construct a tree with N vertices numbered 1 through N, such that for each i = 1,2,...,N, the distance between vertex i and the farthest vertex from it is a_i, assuming that the length of each edge is 1.

Determine whether such a tree exists.

Constraints

2 ≦ N ≦ 100

1 ≦ a_i ≦ N-1

Input

The input is given from Standard Input in the following format:

N
a_1 a_2 ... a_N

Output

If there exists a tree that satisfies the condition, print Possible. Otherwise, print Impossible.

Sample Input 1

5
3 2 2 3 3

Sample Output 1

Possible

The diagram above shows an example of a tree that satisfies the conditions. The red arrows show paths from each vertex to the farthest vertex from it.

Sample Input 2

3
1 1 2

Sample Output 2

Impossible

Sample Input 3

10
1 2 2 2 2 2 2 2 2 2

Sample Output 3

Possible

Sample Input 4

10
1 1 2 2 2 2 2 2 2 2

Sample Output 4

Impossible

Sample Input 5

6
1 1 1 1 1 5

Sample Output 5

Impossible

Sample Input 6

5
4 3 2 3 4

Sample Output 6

Possible

Solution in C++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <cstdio>
#include <cmath>
#include <cstring>
#include <ctime>
#include <climits>
#include <iostream>
#include <algorithm>
#include <set>
#include <map>
#include <queue>
#include <vector>
#include <sstream>
#include <typeinfo>
#include <fstream>

#define DIV 1000000007

using namespace std;
long N;
long a[105];
map<long, long> coun;
vector<pair<long, long> > vec;

int main(){
	cin >> N;
	for(int i = 0; i < N; i++){
		cin >> a[i];
		coun[a[i]]++;
	}
	for(auto c:coun){
		vec.push_back(make_pair(c.first, c.second));
	}
	sort(vec.begin(), vec.end());
	//for(int i = 0; i < vec.size(); i++){
	//	cout << vec[i].first << ":" << vec[i].second << endl;
	//}

	if(N == 1 && a[0] == 0){
		cout << "Possible" << endl;
		return 0;
	}

	if(vec[vec.size() - 1].second < 2){
		cout << "Impossible" << endl;
		return 0;
	}

	if(vec[0].first == 1 && vec[0].second != 1){
		cout << "Impossible" << endl;
		return 0;
	}

	if(vec[0].first > 2){
		cout << "Impossible" << endl;
		return 0;
	}

	for(int i = 1; i < vec.size(); i++){
		if(vec[i-1].first + 1 != vec[i].first){
			cout << "Impossible" << endl;
			return 0;
		}
	}

	cout << "Possible" << endl;
	return 0;
}