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++14 (GCC 5.4.1) (judge verdict: Accepted)

/*
Author: Racer5x
 ***************************************   UNAUTHORISED COPYING OF CODE PROHIBITED   **********************************
*/
 
// #pragma GCC optimize("O3")
// #pragma GCC optimize("Ofast")
// #pragma GCC optimize("unroll-loops")
// #pragma GCC target("sse,sse2,sse3,ssse3,sse4,popcnt,abm,mmx,avx,tune=native")
 
#include<bits/stdc++.h>
#define int         long long
#define double      long double
#define pb          emplace_back
#define pf          emplace_front
#define pii         pair<int,int>
#define vi          vector<int>
#define vii         vector<pii>
#define mi          map<int,int>
#define mii         map<pii,int>
#define all(a)      (a).begin(),(a).end()
#define rall(a)     (a).rbegin(),(a).rend()
#define x           first
#define y           second
#define endl        '\n'
#define hell        998244353 
#define PI          3.141592653589
#define tezz        ios_base::sync_with_stdio(false); cin.tie(0); cout.tie(0);
#define MAX         2000000000000000000
#define M           1000000007
using namespace std;
 
signed main()
{
    tezz
    
    int n, k;
    cin>>n>>k;

    vector<double> v(n), psum(n+1);

    for(int i=0;i<n;i++){
        int x;
        cin>>x;

        double d = (double)(x * (x + 1)) / (2 * x);
        v[i] = d;
    }

    for(int i=0;i<n;i++) psum[i+1] = psum[i] + v[i];

    double mx = 0;
    for(int i=k;i<=n;i++){
        mx = max(mx, psum[i] - psum[i - k]);
    }

    cout<<setprecision(10)<<fixed<<mx;
}