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 <algorithm>
#include <iostream>
using namespace std;

int main() {
  int b[101] = {}, n, a, mini = 100, maxi = 0, cnt = 0;
  cin >> n;
  for (int i = 0; i < n; i++) {
    cin >> a;
    b[a]++;
    if (b[a] >= 3) {
      cnt++;
    }
    mini = min(mini, a);
    maxi = max(maxi, a);
  }

  if (mini != (maxi + 1) / 2) {
    cout << "Impossible" << endl;
    return 0;
  }
  if (maxi % 2 == 0 && b[mini] >= 2) {
    cout << "Impossible" << endl;
    return 0;
  }
  if (n - cnt != maxi + 1) {
    cout << "Impossible" << endl;
    return 0;
  }
  for (int i = mini; i <= maxi; i++) {
    if (!(b[i] >= 2 || i == mini && maxi % 2 == 0)) {
      cout << "Impossible" << endl;
      return 0;
    }
  }
  cout << "Possible" << endl;
}