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.

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.

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.