B757 CREW AVOIDED AN NMAC BY FOLLOWING TCASII RA.

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NEAR MISS WITH B747 RWY 25L LAX. ON FINAL ILS RWY 25L; APCH CTLR ASKED IF WE HAD ARPT IN SIGHT. REPLIED AFFIRMATIVE. CLRED VISUAL RWY 25L. SHORTLY THEREAFTER; APCH CTLR ASKED IF WE HAD A B747 ON DOWNWIND TO BASE FOR THE N SIDE. REPLIED AFFIRMATIVE. TOLD TO MAINTAIN VISUAL WITH B747 FOR THE N SIDE. B747 WAS OBSERVED COMING OFF PERCH FROM DOWNWIND TO BASE. B747 ROLLED WINGS LEVEL ON BASE. POINTED B747 OUT TO FO TO ENSURE HE SAW THE SAME ACFT. FO STATED CORRECTLY THAT B747 SHOULD BE TURNING FINAL RWY 25L LAX JUST INSIDE OF HUNDA. APCH CTLR ASKED IF WE HAD ARPT IN SIGHT. REPLIED AFFIRMATIVE. CLRED VISUAL RWY 25L. APCH CTLR ASKED IF HAD B747 DOWNWIND TO BASE FOR N SIDE. REPLIED AFFIRMATIVE. TOLD TO MAINTAIN VISUAL WITH B747. B747 TURNED BASE AND ROLLED WINGS LEVEL. BASICALLY CONSTANT BEARING DECREASING RANGE WITH VERY SLIGHT AFTWARD TREND. AT THAT POINT; IT APPEARED THAT AN OVERSHOOT WAS OCCURRING. FO STARTED A L TURN AND STARTED LEVELING ACFT. TCASII ALERT 'CLB; CLB.' STARTED CLB AT A 220 DEG HDG. TOLD APCH CTLR THAT WE HAD TO TURN TO AVOID B747 AND COULD NOT SEE HIM. HE HAD PASSED BEHIND AND BELOW OUR WING LINE. GUESS B747 OFFSET ABOUT 1 MI BEFORE HE STARTED TO TURN. WE DEVIATED 1 - 1 1/2 MI S OF CTRLINE FOR AVOIDANCE. TCASII HAD B747 300 FT BELOW AND VERY; VERY CLOSE. SYMBOLOGY FOR B747 WAS TOUCHING TAIL OF PLANE SYMBOL ON TCASII. APCH CTLR ADVISED THAT B747 WAS TURNING AND WE COULD CONTINUE APCH. STARTED SLIGHT TURN TOWARDS FIELD AND SAW B747 WITH DIVERGENT HDG ESTIMATED LESS THAN 1 MI AND 400-500 FT BELOW US. ESTIMATE THAT HE WAS ON CTRLINE FOR RWY 25L OR S OF CTRLINE. SUPPLEMENTAL INFO FROM ACN 459794: I WAS FLYING A VISUAL APCH TO RWY 25L WHEN APCH POINTED OUT A B747 ON A BASE TURN TO RWY 24. WHEN I LOOKED OUT MY WINDOW THE B747 SEEMED TO BE HDG DIRECTLY TOWARDS US; SO I IMMEDIATELY BEGAN A L CLBING TURN AWAY FROM THE B747. ALMOST IMMEDIATELY WE RECEIVED A TCASII RA COMMANDING A CLB. SHORTLY THEREAFTER THE TCASII COMMANDED A 'MONITOR VERT CLB.' WHEN I BEGAN MY L CLB; I LOST VISUAL WITH THE B747. AT APPROX 1 1/2 MI L OF CTRLINE; I BEGAN A TURN BACK TO THE R TO OBTAIN VISUAL WITH THE B747. BACK ON AN APPROX 250 DEG HDG; I SAW THE B747 MAKING A 45 DEG INTERCEPT BACK TOWARDS RWY 24. WE LANDED WITHOUT ANY FURTHER INCIDENT.