I've read about "mate prevails" here on this site, but I couldn't get a clear view on one odd point. I get that the mating move prevails, but if you couldn't do it before your flag falls, you lose.
Let's say it's my opponent's turn. I'm about to mate in 1. They hit the clock, and I run out of time. Then he notices it and claims it while I'm performing checkmate, at the exact same time.
Under FIDE rules, what happens?