J. The Parade
time limit per test
2 seconds
memory limit per test
512 megabytes
standard input
standard output

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.


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 $$$30000$$$.


For each test case, print one integer — the maximum number of soldiers that can participate in the parade.

3 4
7 1 13
1 1
1 3
2 1
1000000000000 1000000000000
4 1
10 2 11 1

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.