Description:
The Berland Army is preparing for a large military parade. It is already decided that the soldiers participating in it will be divided into $$$k$$$ rows, and all rows will contain the same number of soldiers.
Of course, not every arrangement of soldiers into $$$k$$$ rows is suitable. Heights of all soldiers in the same row should not differ by more than $$$1$$$. The height of each soldier is an integer between $$$1$$$ and $$$n$$$.
For each possible height, you know the number of soldiers having this height. To conduct a parade, you have to choose the soldiers participating in it, and then arrange all of the chosen soldiers into $$$k$$$ rows so that both of the following conditions are met:
- each row has the same number of soldiers,
- no row contains a pair of soldiers such that their heights differ by $$$2$$$ or more.
Calculate the maximum number of soldiers who can participate in the parade.
Input Format:
The first line contains one integer $$$t$$$ ($$$1 \le t \le 10000$$$) — the number of test cases. Then the test cases follow.
Each test case begins with a line containing two integers $$$n$$$ and $$$k$$$ ($$$1 \le n \le 30000$$$, $$$1 \le k \le 10^{12}$$$) — the number of different heights of soldiers and the number of rows of soldiers in the parade, respectively.
The second (and final) line of each test case contains $$$n$$$ integers $$$c_1$$$, $$$c_2$$$, ..., $$$c_n$$$ ($$$0 \le c_i \le 10^{12}$$$), where $$$c_i$$$ is the number of soldiers having height $$$i$$$ in the Berland Army.
It is guaranteed that the sum of $$$n$$$ over all test cases does not exceed $$$30000$$$.
Output Format:
For each test case, print one integer — the maximum number of soldiers that can participate in the parade.
Note:
Explanations for the example test cases:
1. the heights of soldiers in the rows can be: $$$[3, 3, 3, 3]$$$, $$$[1, 2, 1, 1]$$$, $$$[1, 1, 1, 1]$$$, $$$[3, 3, 3, 3]$$$ (each list represents a row);
2. all soldiers can march in the same row;
3. $$$33$$$ soldiers with height $$$1$$$ in each of $$$3$$$ rows;
4. all soldiers can march in the same row;
5. all soldiers with height $$$2$$$ and $$$3$$$ can march in the same row.