SYS ERROR AND RESULTING POTENTIAL CONFLICT IN THE OFFING WHEN THE S80 FO TURNS TO THE WRONG HDG ON A MISSED APCH TO ILS RWY 25L AT LAX; CA.

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GAR. AT 1000 FT AGL ON ILS RWY 25L APCH AT LAX; WE WERE ASKED IF WE HAD THE ACFT IN FRONT OF US IN SIGHT. I RESPONDED THAT WE WERE EXECUTING AN INST APCH AND DID NOT. WE WERE THEN GIVEN GAR INSTRUCTIONS OF HDG 250 DEGS; CLB AND MAINTAIN 2000 FT. WE HEARD HDG 280 DEGS INSTEAD. LEVELING AT 2000 FT; WE WERE GIVEN AN IMMEDIATE L TURN TO 220 DEGS. AT THAT TIME WE GOT A TA FROM AN ACFT AT 2-3 O'CLOCK POS AND 300 FT ABOVE US AT ABOUT 5 MI ON TCASII. ONCE ON HDG 220 DEGS; TA RESOLVED AND WE TURNED BACK TO HDG 250 DEGS AS INSTRUCTED AND TOOK VECTORS FOR A NORMAL ILS RWY 25L APCH. FO HAND FLYING DID A GOOD JOB. IN THE FUNK OF TRYING TO FIGURE OUT WHAT HAPPENED TO LOSE SEPARATION; WE WERE DISTR ENOUGH TO MISS THE CORRECT HDG. SUPPLEMENTAL INFO FROM ACN 546527: WE WERE ON FINAL FOR RWY 25L AT LAX AND HAD BEEN CLRED TO LAND BY THE TWR. THE WX AT THE TIME WAS MARGINAL VFR AND WE WERE FOLLOWING A B757. THE TWR CTLR ASKED US IF WE HAD HAD THE PRECEDING B757 IN SIGHT. CAPT SAID WE DIDN'T HAVE HIM IN SIGHT FOR THE INST PORTION OF THE APCH. THE TWR CTLR THEN SAID WHAT I THOUGHT WAS APCH CLRNC CANCELED; MAINTAIN 2000 FT; HDG 280 DEGS. I BEGAN A CLB WITH A SLIGHT BANK TO THE R WHEN THE CAPT POINTED OUT TFC TO OUR R ON THE TCASII. I STARTED TO LEVEL THE WINGS WHEN LAX TWR SAID MAKE AN IMMEDIATE R TURN TO 220 DEGS. I MADE THE TURN AND CONTINUED THE MISSED APCH. NOTHING FURTHER WAS SAID BY ATC.