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 Python (3.8.2) (judge verdict: Wrong Answer)

N = int(input())
A = list(map(int,input().split()))
A.sort()
mn,mx = A[0],A[-1]

from collections import Counter
ctr = Counter(A)

if N==2:
    if A==[1,1]:
        print('Possible')
    else:
        print('Impossible')
    exit()
if ctr[mx] < 2:
    print('Impossible')
    exit()
if (mn == 1 and ctr[mn] > 1) or ctr[mn] > 2:
    print('Impossible')
    exit()

for n in range(mn,mx+1):
    if ctr[n]==0:
        print('Impossible')
        exit()
print('Possible')