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D - Dice in Line

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 $10^{-6}$.

5 3
1 2 2 4 5

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.

4 1
6 6 6 6

3.500000000000

Regardless of which die we choose, the expected value of the number shown is $3.5$.

10 4
17 13 13 12 15 20 10 13 17 11

32.000000000000