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>
#define pb push_back
#define fs first
#define sc second
#define all(x) (x).begin(), (x).end()
#define sz(x) (int)(x).size()
using namespace std;

typedef long long ll;
typedef vector<int> vi;
typedef vector<ll> vll;
typedef vector<double> vd;
typedef pair<int, int> pii;
typedef vector<pii> vpii;

const int N=200;
int c[N], n, m;

int ok() {
    int i;
    for(i=0; i<=m; i++) c[max(i, m-i)]--;
    for(i=1; i<n; i++) {
        if(c[i]<0) return 0;
        if(i<=(m+1)/2 && c[i]>0) return 0;
    }
    return 1;
}

int main() {
    int i, t;
    scanf("%d", &n);
    for(i=0; i<n; i++) {
        scanf("%d", &t);
        c[t]++;
        m=max(m, t);
    }
    printf(ok()? "Possible\n" : "Impossible\n");
    return 0;
}