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)

namespace Test
{
    using System;
    using System.Collections.Generic;
    using System.Linq;
    using System.Text;

    public class Test
    {
        public static void Main(string[] args)
        {
            var test = new Test();
            var input = test.ConvertIntArray(test.ReadInput());
            var n = input[0];
            var k = input[1];
            var array = test.ConvertIntArray(test.ReadInput());
            double ans = 0d;
            var memory = 0;
            var memoryIndex = 0;

            for (var j = 0; j < k; j++)
            {
                memory += array[0 + j];
            }

            for (var i = 1; i <= n - k; i++)
            {
                var temp = 0;
                for (var j = 0; j < k; j++)
                {
                    temp += array[i + j];
                }

                if (memory < temp)
                {
                    memory = temp;
                    memoryIndex = i;
                }
            }

            for (var i = memoryIndex; i < memoryIndex + k; i++)
            {
                double temp = 0d;
                for (var j = 0; j < array[i]; j++)
                {
                    temp += (j + 1) * (1.0 / array[i]);
                }
                ans += temp;
            }
            Console.WriteLine(ans);
        }


        public string[] ReadInput()
        {
            return Console.ReadLine().Split(' ');
        }

        public int[] ConvertIntArray(string[] x)
        {
            var convertedArray = new int[x.Length];
            for (var i = 0; i < x.Length; i++)
            {
                convertedArray[i] = int.Parse(x[i]);
            }

            return convertedArray;
        }

        public double[] ConvertDoubleArray(string[] x)
        {
            var convertedArray = new double[x.Length];
            for (var i = 0; i < x.Length; i++)
            {
                convertedArray[i] = int.Parse(x[i]);
            }

            return convertedArray;
        }

        public long[] ConvertLongArray(string[] x)
        {
            var convertedArray = new long[x.Length];
            for (var i = 0; i < x.Length; i++)
            {
                convertedArray[i] = long.Parse(x[i]);
            }

            return convertedArray;
        }

        public int CountContainWords(string source, string word)
        {
            var count = 0;
            var loopCount = source.Length - (word.Length - 1);
            for (var i = 0; i < loopCount; i++)
            {
                if (source.Substring(i, word.Length) == word)
                {
                    count++;
                }
            }
            return count;
        }
    }
}