1250J · The Parade

Problem: 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.