This is the hard version of the problem. The difference between the versions is that in this version you need to output all the operations that need to be performed. You can hack only if you solved all versions of this problem. You are given (n) arrays, each of which has a length of (m). Let the (j)-th element of the (i)-th array be denoted as (a_{i, j}). It is guaranteed that all (a_{i, j}) are pairwise distinct . In one operation, you can do the following: Choose some integer (i) ((1 \le i \le n)) and an integer (x) ((1 \le x \le 2 \cdot n \cdot m)). For all integers (k) from (i) to (n) in increasing order, do the following: Add the element (x) to the beginning of the (k)-th array. Assign (x) the value of the last element in the (k)-th array. Remove the last element from the (k)-th array. Add the element (x) to the beginning of the (k)-th array. Assign (x) the value of the last element in the (k)-th array. Remove the last element from the (k)-th array. In other words, you can insert an element at the beginning of any array, after which all elements in this and all following arrays are shifted by one to the right. The last element of the last array is removed. You are also given a description of the arrays that need to be obtained after all operations. That is, after performing the operations, the (j)-th element of the (i)-th array should be equal to (b_{i, j}). It is guaranteed that all (b_{i, j}) are pairwise distinct . Determine the minimum number of operations that need to be performed to obtain the desired arrays, and also output the sequence of all operations itself. Each test consists of multiple test cases. The first line contains a single integer (t) ((1 \le t \le 10^4)) — the number of test cases. The description of the test cases follows. The first line of each test case contains two integers (n) and (m) ($$$1 \le n, m \le