Prove that the first list contains numbers from k to 2k and second list contains numbers from 1 to k.

There are two lists that contain integers from 1 to 2k such that the first list contains k randomly chosen integers from the range 1 - 2k and the second list contains the remaining numbers.
For Example: A = {8, 6, 13, 14, 4, 11, 12} and B = {1, 2, 3, 5, 7, 9, 10} when k = 7.
Now arrange the first list such that it is in ascending order and the second list such that it is in descending order.
Same Example: A = {4, 6, 8, 11, 12, 13, 14} and B = {10, 9, 7, 5, 3, 2, 1}.
Now check the elements in the same position in both lists and make a swap such that the first list contains the greater number while the second list contains the smaller number.
Same Example: A = {10,9,8,11,12,13,14} and B = {1,6,7,5,3,2,1}
Prove that the first list contains numbers from k to 2k and second list contains numbers from 1 to k.

It will be really helpful if I can get some directions on how to do this question because a similar question of sort was asked but I don't think that the answer to that question was similar. Is difference of two consecutive sums of consecutive integers (of the same length) always square?
I have tried it several times and looked for any similar problem but couldn't find them. I know this might be very obvious but please help me out and If there is a similar problem with the solution I would be interested about reading up on it. Thanks.

Hint: Take your example. Once $A$ and $B$ are sorted (but before the swapping is done), they have this overall structure: $$ A=\{(\le 7), (\le 7), (>7), (>7), (>7), (>7), (>7)\}, \\ B=\{(>7), (>7), (\le 7), (\le 7), (\le 7), (\le 7), (\le 7)\}. $$ The thing to note is that they both start with the same number of elements (in this case $2$ elements) that belong in the other set and will be swapped, and after that all elements (i.e. element number $3$ on onwards) are already in their proper place and will not be swapped.

Try to generalise this.