This is the hard version of the problem. The difference between the versions is that in this version, the array (a) is arbitrary. You can hack only if you solved all versions of this problem. Given an array (a) of size (n), and a parameter (k), an array (b) is called cool if the following conditions hold: For each (i) from (k) to (n), the array (a_{i-k+1},a_{i-k+2},\ldots,a_i) is a rearrangement of (b_{i-k+1},b_{i-k+2},\ldots,b_i). You are given two arrays (a) and (b) of size (n), and an integer (k). The array (a) only contains integers from (1) to (n). The array (b) only contains integers from (1) to (n), and (-1). Determine if it is possible to replace all (-1) in (b) with an integer from (1) to (n), such that (b) is cool with respect to (k). Each test contains multiple test cases. The first line contains the number of test cases (t) ((1 \le t \le 10^4)). The description of the test cases follows. The first line of each test case contains two integers (n) and (k) ((1 \leq k \leq n \leq 2\cdot 10^5)). The second line of each test case contains (n) integers (a_1,a_2,\ldots,a_n) ((1 \leq a_i \leq n)). The third line of each test case contains (n) integers (b_1,b_2,\ldots,b_n) ((1 \leq b_i \leq n) or (b_i=-1)). It is guaranteed that the sum of (n) across all test cases does not exceed (2\cdot 10^5). For each test case, output YES if possible, and NO otherwise. You may print each letter in either uppercase or lowercase. In the first test case, we have (k=5). The only subarray of size (5) is (1,5). We can see that (b) is a rearrangement of (a), so the answer is YES . In the second test case, we can have (b=2,1,2,1,2). We can see that each window of size (2) in both (a) and (b) are either (1,2) or (2,1), which are rearrangements of each other, s