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 static System.Console;
using System.Collections.Generic;
using System.Linq;
using System.Text;
using System.Threading.Tasks;
using System.Threading;

namespace Atcoder_Answer
{
    class Program
    {
        const int mod = 1000000007;

        static void Main(string[] args)
        {
            string[] Reader = ReadLine().Split(' ');
            int N = int.Parse(Reader[0]);
            int K = int.Parse(Reader[1]);
            Reader = ReadLine().Split(' ');
            float[] Ps = Reader.Select(float.Parse).ToArray();
            float max = 0;
            
            for (int loop = 0; loop < N - K + 1; loop++) 
            {
                float[] PkSSource = new float[K];
                Array.Copy(Ps, loop, PkSSource, 0, K);
                float PkS = PkSSource.Sum();
                if (((PkS + K) * (PkS - K + 1) / 2) / (PkS - K + 1) > max) max = ((PkS + K) * (PkS - K + 1) / 2) / (PkS - K + 1);
            }

            WriteLine(max);
        }

    }

    public static class Extension
    {
        #region 拡張:bool
        #region キャスト関連
        public static int ToInt(this bool Bool)
        {
            if (Bool) return 1;
            else return 0;
        }

        public static string ToString(this bool Bool, string TrueStr, string FalseStr)
        {
            if (Bool) return TrueStr;
            else return FalseStr;
        }
        #endregion
        #endregion

        #region 拡張:int
        #region キャスト関連
        public static char ToAlphaBets(this int Base) => (char)Base;
        #endregion
        #endregion

        #region 拡張:char
        #region 変換関連
        public static char NextAlpha(this char Char) => (char)((Char) + 1); //次の文字を取得
        #endregion
        #endregion

        #region 拡張:int[]
        public static long Sum(this int[] Array)
        {
            int Ret = 0;
            for (int loop = 0; loop < Array.Length; loop++) Ret += Array[loop];
            return Ret;
        }
        #endregion

        #region 拡張: string
        public static string Multi(this string THIS, int Value) 
        {
            string RetValue = "";

            for (int loop = 0; loop < Value; loop++) RetValue += THIS;

            return RetValue;
        }
        #endregion
    }
}