Problem:
Score : 400 points

Problem Statement

We have N dice arranged in a line from left to right. The i-th die from the left shows p_i numbers from 1 to p_i with equal probability when thrown.

We will choose K adjacent dice, throw each of them independently, and compute the sum of the numbers shown. Find the maximum possible value of the expected value of this sum.

Constraints

1 ≤ K ≤ N ≤ 200000

1 ≤ p_i ≤ 1000

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K
p_1 ... p_N

Output

Print the maximum possible value of the expected value of the sum of the numbers shown.

Your output will be considered correct when its absolute or relative error from our answer is at most 10^{-6}.

Sample Input 1

5 3
1 2 2 4 5

Sample Output 1

7.000000000000

When we throw the third, fourth, and fifth dice from the left, the expected value of the sum of the numbers shown is 7. This is the maximum value we can achieve.

Sample Input 2

4 1
6 6 6 6

Sample Output 2

3.500000000000

Regardless of which die we choose, the expected value of the number shown is 3.5.

Sample Input 3

10 4
17 13 13 12 15 20 10 13 17 11

Sample Output 3

32.000000000000

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

program main
    implicit none
    integer(4):: n,k,i
    integer(4),allocatable:: p(:)
    real(8),allocatable:: s(:)
    real(8):: ans
    read*, n, k
    allocate(p(n))
    read*, p(:)

    call merge_sort(p(:),1,n)

    allocate(s(p(1)))
    s(1) = 1d0
    do i=2, p(1)
        s(i) = s(i-1)+dble(i)
    end do

    do i=1,p(1)
        s(i) = s(i)/dble(i)
    end do


    ans = 0

    do i=1,k
        ans = ans + s(p(i))
    end do
    print*, ans

contains
recursive subroutine merge_sort(ar,fst,lst)
    integer(4):: ar(:),fst,lst
    integer(4):: tmp, mdl
    if(lst-fst < 2) then
        if (ar(fst) < ar(lst)) then
            tmp = ar(fst)
            ar(fst) = ar(lst)
            ar(lst) = tmp
        end if
        return
    end if

    mdl = (fst+lst)/2
    call merge_sort(ar(:), fst, mdl)
    call merge_sort(ar(:), mdl+1, lst)
    call merge(ar(:), fst, mdl, lst)

end subroutine

subroutine merge(ar,fst,mdl,lst)
    integer(4)::ar(:),fst,mdl,lst
    integer(4):: l,r,t
    integer(4),pointer:: tmp(:)

    allocate(tmp(lst-fst+1))
    l=fst
    r=mdl+1
    t=1
    do while(l <= mdl .and. r <= lst)
        if (ar(l) > ar(r)) then
            tmp(t) = ar(l)
            l=l+1
        else
            tmp(t) = ar(r)
            r=r+1
        end if
        t=t+1
    end do
    
    if (r > lst) then
        tmp(t:) = ar(l:mdl)
    else
        tmp(t:) = ar(r:lst)
    end if

    ar(fst:lst) = tmp(:)
    deallocate(tmp)
end subroutine
end program main