Using your Head is Permitted

After trying for a while, unsuccessfully, to come up with such a protocol, Abraham and Benjamin realize that there is a loop-hole that they can, potentially, employ: There is a computer, which we'll nickname "Computer O", that both Abraham and Benjamin have access to through communication channels that are unlimited in capacity. (They can communicate with Computer O in addition to being able to send and receive O(1) bits in each direction through the direct communication line between them.)

Unfortunately, Computer O is programmed to do only one thing: it accepts an order permutation, A, in [1..n] from Abraham and an order permutation, B, in [1..n] from Benjamin, after which it computes whether the combined permutation AοB has exactly one cycle. This result, a Boolean, is then transferred back to both Abraham and Benjamin, and following this Computer O's software terminates permanently.

Abraham and Benjamin cannot re-program Computer O to perform any other task. They cannot see each other's inputs to the program. They cannot even reprogram Computer O to work on permutations of any size other than n. Furthermore, they cannot run Computer O's program more than once: once a pair of permutations is analyzed and a Boolean is returned, the computer halts and cannot be queried again.

The question: Can Abraham and Benjamin figure out whether the two subsets they received have any commonalities, taking into account that they can enlist the help of Computer O? If not - show why; if so - show how.

List of solvers: