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;
using System.Collections.Generic;
using System.Linq;

namespace ForATCODER
{
    public class Program
    {
        ///template
        ///
        /// String[] s = Console.ReadLine().Split(' ');
        ///   long[] nk =  s2larray(Console.ReadLine().Split(' '));
        ///   

        public static void Main(string[] args)
        {
          long[] nk =  s2larray(Console.ReadLine().Split(' '));
            long[] pn = s2larray(Console.ReadLine().Split(' '));
            double pmax = 0;
            for(long i = 0; i <= nk[0];i++)
            {
                long first = i - nk[1];
                long last = first + nk[1];
                if(first < 0)
                {
                    last -= first;
                    first = 0;
                    i = last;
                }
                double p = 0;
                for(long k = first;k < last;k++)
                {
                    long me = pn[k];
                    double p_ = summ(me) / (double)me;
                    p += p_;
                }
                if (pmax < p) pmax = p;
            }
            RESULT(pmax);
        }
      static   long summ(long s)
        {
            if (s == 1) return 1;
            return s + summ(s-1);
        }
        public static bool checka(int[] a, int m)
        {
            int back = 1;
            for (int i = 0; i < a.Length; i++)
            {
                if (a[i] >= 1 && a[i] <= m)
                {
                    if (a[i] >= back)
                    {
                        back = a[i];
                    }
                    else
                    {
                        return false;
                    }
                }
                else
                {
                    return false;
                }
            }
            return true;
        }
        public static ulong pow(int a, int b)
        {
            ulong res = 1;
            for (int i = 0; i < b; i++)
            {
                res *= (ulong)a;
            }
            return res;
        }
        static int s2i(String s)
        {
            return Convert.ToInt32(s);
        }
        static long s2l(String s)
        {
            return Convert.ToInt64(s);
        }
        static void RESULT(double res)
        {
            Console.WriteLine(res);
        }
        static void RESULT(long res)
        {
            Console.WriteLine(res);
        }
        static void RESULT(ulong res)
        {
            Console.WriteLine(res);
        }
        static void RESULT(int res)
        {
            Console.WriteLine(res);
        }
        static void RESULT(String res)
        {
            Console.WriteLine(res);
        }
        static String[] ReadLineAll(int count)
        {
            String[] res = new String[count];
            for (int i = 0; i < count; i++)
            {
                res[i] = Console.ReadLine();
            }
            return res;
        }

        static int[] s2iarray(String[] src)
        {
            int len = src.Length;
            int[] res = new int[len];
            for (int i = 0; i < len; i++)
            {
                res[i] = Convert.ToInt32(src[i]);
            }
            return res;
        }
        static long[] s2larray(String[] src)
        {
            int len = src.Length;
            long[] res = new long[len];
            for (int i = 0; i < len; i++)
            {
                res[i] = Convert.ToInt64(src[i]);
            }
            return res;
        }
        static ulong[] s2larray2(String[] src)
        {
            int len = src.Length;
            ulong[] res = new ulong[len];
            for (int i = 0; i < len; i++)
            {
                res[i] = Convert.ToUInt64(src[i]);
            }
            return res;
        }

        static long C(int n, int r)
        {
            if (n <= 1) return 0;
            if (r > n - r) r = n - r; // because C(n, r) == C(n, n - r)
            long ans = 1;
            int i;

            for (i = 1; i <= r; i++)
            {
                ans *= n - r + i;
                ans /= i;
            }

            return ans;
        }
        static long C(long n, long r)
        {
            if (n <= 1) return 0;
            if (r > n - r) r = n - r; // because C(n, r) == C(n, n - r)
            long ans = 1;
            int i;

            for (i = 1; i <= r; i++)
            {
                ans *= n - r + i;
                ans /= i;
            }

            return ans;
        }
        static ulong C(ulong n, ulong r)
        {
            if (n <= 1) return 0;
            if (r > n - r) r = n - r; // because C(n, r) == C(n, n - r)
            ulong ans = 1;
            ulong i;

            for (i = 1; i <= r; i++)
            {
                ans *= n - r + i;
                ans /= i;
            }

            return ans;
        }
    }

}