Hello! My bathroom faucet has an insert aerator. We attempted to remove it in order to remove the chalk in the faucet, and broke the bottom part off as a result. The keys we have to remove the aerator are now not working, and we can't quite get it out.

Does anybody have any suggestions? We've tried using thin pliers for electronics to grab it and twist, but we can't get a good enough grip. A few photos attached: the remaining part inside the faucet, the broken off bottom part, and the keys that we have.