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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import           Control.Monad
import           Control.Monad.ST
import qualified Data.Array                    as A
import qualified Data.Array.Unboxed            as AU
import qualified Data.ByteString.Char8         as BS
import           Data.Char
import           Data.Int
import qualified Data.IntMap                   as M
import           Data.List
import           Data.Maybe
import           Data.Ord
import           Data.Time
import qualified Data.Vector                   as V
import qualified Data.Vector.Mutable           as VM
import qualified Data.Vector.Unboxed           as VU
import qualified Data.Vector.Unboxed.Mutable   as VUM

readLine :: Read a => IO [a]
readLine = map (read . BS.unpack) . BS.words <$> BS.getLine

readLines :: Read a => Int -> IO [[a]]
readLines n = replicateM n readLine

toPair :: [a] -> (a, a)
toPair [x, y] = (x, y)

main :: IO ()
main = do
  [n, k] <- readLine
  p      <- readLine
  print $ solve n k p

solve :: Int -> Int -> [Int] -> Double
solve n k p = fromIntegral (p' + k) / 2
 where
  p' = maximum $ map (sum . take k) $ filter (\x -> length x >= k) $ tails p
  f :: [Double] -> Double -> Double
  f []       acc = acc
  f (a : as) acc = f as (acc + (1 + a) / 2)