D. Decidophobia

Source: Codeforces 2240D
Contest: Codeforces Round 1105 (Div. 2)
Time limit: 2 seconds
Memory limit: 256 megabytes

Statement

There are $n$ people attending a round-table party, numbered $1, 2, 3, \ldots, n$ in clockwise order. You have prepared some gifts to distribute among them.

Each person $i$ has a weight $a_i$ and a common field of view $d$. The field of view for person $i$ consists of the $d$ people sitting clockwise and the $d$ people sitting counter-clockwise from them (a total of $2d$ people excluding person $i$).

The happiness gained by person $i$ is determined by the following rules:

• If person $i$ receives a gift, and there are $x$ people within their field of view who did not receive a gift, they gain $x \cdot a_i$ happiness.
• If person $i$ does not receive a gift, and there are $x$ people within their field of view who received a gift, they incur $-x \cdot a_i$ happiness.

You want to maximize the total happiness of all $n$ people combined. Find this maximum value.

Input

Each test contains multiple test cases. The first line contains the number of test cases $t$ ($1 \le t \le 10^4$). The description of the test cases follows.

The first line of each test case contains two integers $n$ and $d$ ($3 \le n \le 2 \cdot 10^5$, $1 \le d \lt \frac{n}{2}$).

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($1 \le a_i \le 10^8$), where $a_i$ is the weight of the $i$-th person.

It is guaranteed that the sum of $n$ over all test cases does not exceed $10^6$.

Output

For each test case, output a single integer representing the maximum total happiness.

Note

In the first test case, there are 3 people sitting in a circle. For each person $i$, the field of view is $d=1$, which includes the 1 person sitting clockwise and the 1 person sitting counter-clockwise from them. If person 2 receives the gift, they gain happiness from 2 neighbors who did not receive it, resulting in $2 \cdot a_2 = 2 \cdot 2 = 4$. However, person 1 and person 3 incur loss because they did not receive a gift but have a neighbor who did. After calculating all possibilities, the maximum total happiness is 3.

In the second test case, $n=5, d=1$. The best solution is to give gifts to persons 2, 3, and 5, which can achieve the maximum total happiness value of 15.

Examples

5
3 1
1 2 3
5 1
1 4 5 2 6
6 2
1 1 4 5 1 4
10 2
230 24 3 42 432 234 934 2389 333 444
3 1
100000000 100000000 100000000

3
15
26
8590
0