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 <iostream>
#include <vector>

using namespace std;

void failure () {
  cout << "Impossible" << endl;
  exit(0);
}

const int MAX_N = 105;

int atds [MAX_N]; // # of vertices with such farthest dist

int main () {
  ios::sync_with_stdio(false);

  int n;
  cin >> n;

  int diam = 0;
  for (int i = 0; i < n; i++) {
    int x;
    cin >> x;

    atds[x]++;
    diam = max(diam, x);
  }

  int blw, abv;
  if (diam % 2 == 0) {
    if (atds[diam / 2] != 1) failure();
    blw = diam / 2 - 1;
    abv = diam / 2 + 1;
  } else {
    blw = diam / 2;
    abv = (diam + 1) / 2;
  }

  for (int i = 0; i <= blw; i++) {
    if (atds[i] > 0) failure();
  }
  for (int i = abv; i <= diam; i++) {
    if (atds[i] < 2) failure();
  }

  cout << "Possible" << endl;
}