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<set>
using namespace std;

int main(){
    int n;
    cin >> n;
    multiset<int> s;
    for(int i = 0; i < n; i++){
        int x;  cin >> x;
        s.insert(x);
    }
    int diam = *s.rbegin();
    bool even = diam%2==0;
    int to = diam/2+1;
    while(diam >= to){
        bool valid = s.size() >= 2;
        for(int i = 0; i < 2; i++){
            if(valid && s.count(diam))      s.erase(s.find(diam));
            else                            valid = false;
        }
        if(!valid){
            cout << "Impossible" << endl;
            return 0;
        }
        diam--;
    }
    if(even){
        if(s.size() == 0 || s.count(to-1) == 0){
            cout << "Impossible" << endl;
            return 0;
        }else{
            s.erase(s.find(to-1));
        }
    }
    cout << (s.size() == 0 || *s.begin() >= to ? "Possible" : "Impossible") << endl;
    return 0;
}