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If you do the trip to GIG first and use AA miles for the GIG-MIA return flight, then you can make MIA your gateway stopover city, and continue on -- at a later date -- to another North American city (like LAX or HNL) for no additional miles on the same award. But all travel on the award must be completed within one year of ticketing date.

If you go to SYD first and are able to fly back to HNL or LAX, you could get an AA award HNL/LAX-MIA (stopover) - GIG.