CF1250J.The Parade

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.

伯兰德军队正在为一场大型军事阅兵式做准备。目前已确定，参加阅兵式的士兵将被分为 $k$ 行，且每行士兵人数相同。

当然，并非所有将士兵划分为 $k$ 行的方式都合适。同一行中所有士兵的身高之差不能超过 $1$。每位士兵的身高是一个介于 $1$ 到 $n$ 之间的整数。

对于每个可能的身高值，你已知具有该身高的士兵人数。为举行阅兵式，你需要先选出参与阅兵的士兵，再将所有选中的士兵安排成 $k$ 行，使得同时满足以下两个条件：

• 每行士兵人数相同；
• 任意一行中均不存在两名士兵，其身高之差为 $2$ 或更大。

请计算最多可有多少名士兵参加阅兵式。

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 $30000$.

第一行包含一个整数 $t$（$1 \le t \le 10000$），表示测试用例的数量。随后是各测试用例。

每个测试用例以一行开始，该行包含两个整数 $n$ 和 $k$（$1 \le n \le 30000$，$1 \le k \le 10^{12}$），分别表示士兵的不同身高种类数以及阅兵队伍的行数。

每个测试用例的第二行（也是最后一行）包含 $n$ 个整数 $c_1$, $c_2$, ..., $c_n$（$0 \le c_i \le 10^{12}$），其中 $c_i$ 表示贝兰德军队中身高为 $i$ 的士兵人数。

保证所有测试用例的 $n$ 之和不超过 $30000$。

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

对于每个测试用例，输出一个整数——即最多可以参加阅兵的士兵人数。

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.

示例测试用例的解释：

1. 各行士兵的身高可以为：$[3, 3, 3, 3]$、$[1, 2, 1, 1]$、$[1, 1, 1, 1]$、$[3, 3, 3, 3]$（每个列表代表一行）；
2. 所有士兵均可排在同一行中；
3. 每行各有 $33$ 名身高为 $1$ 的士兵，共 $3$ 行；
4. 所有士兵均可排在同一行中；
5. 所有身高为 $2$ 和 $3$ 的士兵均可排在同一行中。