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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
class Program
{
    static void Main(string[] args)
    {
            // 入力
            string[] sS = System.Console.ReadLine().Trim().Split(' ');
            int iN = int.Parse(sS[0]);
            int iK = int.Parse(sS[1]);
            sS = System.Console.ReadLine().Trim().Split(' ');
            List<int> lstInp = new List<int>();
            for (int i = 0; i < sS.Length; i++)
            {
                lstInp.Add(int.Parse(sS[i]));
            }

            System.Diagnostics.Stopwatch sw = new System.Diagnostics.Stopwatch();
            sw.Start();

            // 解析
            List<ClsSum> lstRet = new List<ClsSum>();
            for (int i = 0; i <= lstInp.Count - iK; i++)
            {
                int iSum = 0;
                for (int j = i; j < i + iK; j++) { iSum += lstInp[j]; }
                lstRet.Add(new ClsSum() { iS = iSum, iX = i });
            }
            lstRet.Sort(delegate (ClsSum mca1, ClsSum mca2) { return mca2.iS - mca1.iS; });     // 合計の降順
            decimal iOut = 0M;
            for (int j = lstRet[0].iX; j < lstRet[0].iX + iK; j++) { iOut += (decimal)lstInp[j] / 2M + 0.5M; }

            // 出力
            System.Console.WriteLine("{0}", iOut.ToString().Trim());

            sw.Stop();
#if DEBUG
            Console.WriteLine();
            Console.WriteLine("{0}", sw.Elapsed.ToString());
            Console.WriteLine("続行するには何かキーを押してください．．．");
            Console.ReadKey();
#endif
    }

        // Grp情報
        private class ClsSum
        {
            public int     iS { get; set; } = 0;                // 合計
            public int     iX { get; set; } = 0;                // 位置Index
        }

}