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: Accepted)

#include <bits/stdc++.h>
using namespace std;

int main() {
    int N;
    cin >> N;
    map<int, int> m;
    int mi = 1000, ma = 0;
    for(int i = 0; i < N; ++i) {
        int a;
        cin >> a;
        m[a] += 1;
        mi = min(mi, a);
        ma = max(ma, a);
    }
    if(N == 1) {
        cout << "Impossible" << endl;
        return 0;
    }

    bool ok = (m[mi] >= 2 && ma & 1) || (m[mi] == 1 && ma % 2 == 0);
    ok &= (ma + 1) / 2 == mi && ma <= N - 1 && m[mi] <= 2;
    int prev = mi - 1;
    for(auto& p : m) {
        if(p.first != prev + 1) {
            ok = false;
        }
        prev = p.first;
        if(p.first == mi) {
            continue;
        }
        ok &= p.second >= 2;
    }

    if(ok) {
        cout << "Possible" << endl;
    } else {
        cout << "Impossible" << endl;
    }
}