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 Fortran (gfortran v4.8.4) (judge verdict: Wrong Answer)

program tree_restoring
  implicit none
  integer :: n, a(100), i
  a = 0
  read(*,*) n
  if (n.eq.2) then
    write(*,'(a)') "Possible"
    stop
  end if
  read(*,*) a(1:n)
  call quick_sort(a(1:n))
  do i = 2, n
    if (a(i).ne.a(i-1).and.a(i).ne.a(i-1)+1) then
      write(*,'(a)') "Impossible"
      stop
    end if
  end do
  if (a(n).ne.a(n-1)) then
    write(*,'(a)') "Impossible"
    stop
  end if
  if (a(1).ne.(a(n)+1)/2) then
    write(*,'(a)') "Impossible"
    stop
  end if
  if (mod(a(n),2).eq.0) then
    if (a(1).eq.a(2)) then
      write(*,'(a)') "Impossible"
      stop
    end if
  else
    if (a(1).ne.a(2)) then
      write(*,'(a)') "Impossible"
      stop
    end if
  end if
  write(*,'(a)') "Possible"
  stop
contains
  recursive subroutine quick_sort(a)
    implicit none
    integer, intent(inout) :: a(:)
    integer :: n, l, r, p, c
    n = size(a)
    l = 1
    r = n
    p = a((l+r)/2)+a(l)+a(r)-max(a((l+r)/2),a(l),a(r))-min(a((l+r)/2),a(l),a(r))
    do
      do while (a(l).lt.p)
        l = l+1
      end do
      do while (a(r).gt.p)
        r = r-1
      end do
      if (l.ge.r) exit
      c = a(l)
      a(l) = a(r)
      a(r) = c
      l = l+1
      r = r-1
    end do
    if (l.gt.2) call quick_sort(a(1:l-1))
    if (n.gt.r+1) call quick_sort(a(r+1:n))
    return
  end subroutine quick_sort
end program tree_restoring