Square-free division (hard version)

数组 $a$ 由 $n$ 个正整数构成。你需要将它们分割成最小数量的连续子段，使得每一个子段中的任意两个数（不同位置）的乘积不为完全平方数。 除此之外，你被允许在分割之前进行最多 $k$ 次修改操作。 在一次修改操作中，你可以选择数组中的某个位置的数，将该位置的数变为任意正整数。 请问连续子段的最小数量是多少（在最多 $k$ 次操作后）？

This is the hard version of the problem. The only difference is that in this version $ 0 \leq k \leq 20 $ . There is an array $ a_1, a_2, \ldots, a_n $ of $ n $ positive integers. You should divide it into a minimal number of continuous segments, such that in each segment there are no two numbers (on different positions), whose product is a perfect square. Moreover, it is allowed to do at most $ k $ such operations before the division: choose a number in the array and change its value to any positive integer. What is the minimum number of continuous segments you should use if you will make changes optimally?

The first line contains a single integer $ t $ $ (1 \le t \le 1000) $ — the number of test cases. The first line of each test case contains two integers $ n $ , $ k $ ( $ 1 \le n \le 2 \cdot 10^5 $ , $ 0 \leq k \leq 20 $ ). The second line of each test case contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le 10^7 $ ). It's guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

For each test case print a single integer — the answer to the problem.

3
5 2
18 6 2 4 1
11 4
6 2 2 8 9 1 3 6 3 9 7
1 0
1

1
2
1

In the first test case it is possible to change the array this way: $ [\underline{3}, 6, 2, 4, \underline{5}] $ (changed elements are underlined). After that the array does not need to be divided, so the answer is $ 1 $ . In the second test case it is possible to change the array this way: $ [6, 2, \underline{3}, 8, 9, \underline{5}, 3, 6, \underline{10}, \underline{11}, 7] $ . After that such division is optimal: - $ [6, 2, 3] $ - $ [8, 9, 5, 3, 6, 10, 11, 7] $