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 <fstream>
using namespace std;
const int nmax=105;
int n,a[nmax],ap[nmax],req[nmax];
int ok,i,mx;
int main()
{
    freopen("data.in","r",stdin);
    cin>>n;
    for(i=1;i<=n;i++)
        {cin>>a[i];mx=max(mx,a[i]);ap[a[i]]++;}
    if(mx==1&&n!=2)
    {
        cout<<"Impossible";
        return 0;
    }
    for(i=(mx+1)/2;i<=mx;i++)
        req[i]=2;
    if(mx%2==0)
        req[mx/2]=1;
    ok=1;
    for(i=1;i<=n;i++)
        if(ap[i]<req[i])
             ok=0;
    for(i=1;i<mx/2;i++)
        if(ap[i])
            ok=0;
    if(mx%2==0&&ap[mx/2]>1)
        ok=0;
    if(ok) cout<<"Possible";
    else cout<<"Impossible";
    return 0;
}