CREW MISSES A TURN AND CROSSES ACTIVE RWY WITHOUT CLRNC.

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TAXIING OUTBOUND AT MIA; I MISSED A TURN TO GO TO THE CORRECT RWY. I CROSSED A RWY; WHICH I BELIEVE WAS ACTIVE; AS A RESULT. NO CONFLICT OCCURRED AND WE WERE CLRED TO ANOTHER RWY FOR TKOF. ONE REASON FOR MY ERROR WAS THAT I HAD CALLED FOR THE TAXI CHKLIST AND WAS HELPING ACCOMPLISH IT AT A BAD TIME. I SHOULD HAVE FOCUSED MY ATTN ON WHERE WE WERE GOING FIRST. SUPPLEMENTAL INFO FROM ACN 448157: UPON CONTACTING GND CTL; WE WERE TOLD TO TAXI TO RWY 13. FROM THE CONCOURSE; THIS ENTAILS TAXIING DOWN TXWY Q OR TXWY P TO RWY 13. TXWY T INTERSECTS TXWYS Q AND P; AND PARALLELS RWYS 9/27L. AS WE BEGAN TO MOVE; I ASKED; 'ARE YOU FAMILIAR?' REFERRING TO THE TAXI OUT. THE CAPT INDICATED HE WAS; AND I BEGAN TO BUSY MYSELF WITH CHKLISTS. THE NEXT THING I KNEW GND CTL WAS SAYING; 'ACR X; YOU'RE GOING THE WRONG WAY. UH; OK; TAXI TO RWY 9R.' WE HAD PREVIOUSLY COMMENTED ON A B747 THAT WE FOLLOWED OUT FROM THE RAMP AREA AND I THINK WE JUST GOT IN LINE BEHIND HIM. WHEN GND CALLED US; I LOOKED UP TO FIND US ACTUALLY XING RWYS 13/31 ON TXWY T. THE FAA JUMP SEATER QUERIED US ABOUT WHICH RWY WE WERE GOING TO; AND WE REPLIED THAT WE HAD BEEN REASSIGNED RWY 9R; WHICH WE HAD; BUT ONLY AFTER WE HAD TAXIED ONTO AN ACTIVE RWY 13. I DON'T THINK THAT HE NOTICED THE TRANSGRESSION. HE HAD NOTICED WE WEREN'T TAXIING TO RWY 13 ANYMORE. HE HAD INDICATED EARLIER THAT HE HAD BEEN UP ALL NIGHT GIVING A SIMULATOR CHK AND WAS PRETTY TIRED. THE REST OF THE TRIP PASSED UNEVENTFULLY (I FLEW THE LEG) AND THE FAA GUY COMPLIMENTED US ON OUR PROFESSIONALISM. ACTUALLY; I'M MORTIFIED BY THE FACT THAT WE TAXIED ACROSS THE RWY; ESPECIALLY WITH THE FAA IN THE JUMP SEAT. THIS COULD HAVE BEEN A DISASTER; BOTH FROM THE FLAMING WRECKAGE STANDPOINT AND/OR A VIOLATIONS STANDPOINT. I LEARNED MY LESSON: ALWAYS WATCH AND CONFIRM WHAT THE OTHER PLT IS DOING EVEN IF HE SAYS HE KNOWS WHERE HE'S GOING; AND PAY ATTN AT ALL TIMES WHILE TAXIING. I FAILED TO CONFIRM THE R TURN ONTO TXWY Q; AND LOOK WHAT COULD HAVE HAPPENED.