Never mind strictly, wasn't someone grumbling on here the other day about posts that had nothing at all to do with BA? ;-)

Anyway...

Are you set on just this specific combination of flights?

I assume you've never tried to use online travel sites (opodo, expedia, cheapoair) or sites that give you potential routings and prices (google flights, dohop.com) - they will be the ones to give you prices for these sorts of more complex routings. I'll comment that if you want to travel on these specific flights under these codes, it's likely to be tricky, as you'd need fares that allow you to combine the different carriers (Cathay, AA). And for tricky, read expensive.

But from looking at, eg cheapoair, I see that you can actually get the entire journey on AA (including the AA code on the CX flight, which brings the price down to sensible levels), and on google flights it gives a quote for the same itinerary but using the Cayman Airways flight at the start (which is a bit more expensive). In pounds, I'm seeing £1300 and £1500 respectively for the two options, but you'll be quoted something different in a different area of sale, probably...

But try those options first, would be my recommendation! Good luck!