ARTCC VECTORED LGT TO MISS UNKNOWN VFR TFC; RESULT WAS POSSIBLE NMAC.

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AFTER TKOF SNA WE WENT TO COAST DEP; THEN HANDED OFF TO COAST APCH. COAST APCH ASKED FOR HIGH RATE OF CLB FOR XING TFC AT 10000 FT. WHEN LEAVING 10000 FT WE WERE HANDED OFF TO ZLA; CLRED TO FL230. AT ABOUT 13500 FT CTLR ASKED US TO LEVEL AT 14000 FT FOR TFC. (WE WERE STILL AT A HIGH RATE OF CLB (3000 FPM) FROM PREVIOUS CTLR'S REQUEST; AND GENERALLY LIKE TO GET TO FL180 QUICKLY DUE TO HIGH DENSITY TFC IN THIS AREA.) ALMOST SIMULTANEOUSLY AFTER 14000 FT REQUEST CTLR ADVISED 'IF YOU CANNOT MEET 14000 FT RESTRICTION TURN L 30 DEGS.' BY THIS TIME CAPT HAD OVERRIDDEN AUTOPLT; AS IT WAS OBVIOUS IT WOULD NOT BE ABLE TO HOLD THE 14000 FT. HE ROLLED L AND STOPPED CLB AT ABOUT 14400. HE COMMENCED DSCNT AND CTLR TOLD US TO TURN ANOTHER 30 DEGS L; AND COMMENCE DSCNT. AT THIS TIME TCASII STARTED WARNING ABOUT APCHING TFC AND TELLING US TO DSND. CTLR CONTINUED ISSUING INSTRUCTIONS AND WE ENDED UP AT ABOUT 13500 WITH ABOUT 60-70 DEGS OF DIRECTIONAL CHANGE. WE NEVER SAW THE ACFT; BUT ACFT APPEARED ON TCASII TO HAVE PASSED WITHIN 100-200 OF OUR ACFT. AT ONE TIME; CTLR HAD ADVISED HE THOUGHT IT WOULD PASS TO OUR R. FLT CONTINUED TO SLC. ON REFLECTION AND AFTER DISCUSSING IT WITH CAPT WE FEEL WE DIDN'T GET AN EARLIER TCASII WARNING BECAUSE OF OUR HIGH RATE OF CLB TO BEGIN WITH. TCASII; THINKING WE WOULD BE ABOVE APCHING TFC WHEN CTLR HAD US LEVEL OFF; BECAME MORE OF A CONFLICT AND WE BEGAN TO GET WARNINGS. I BELIEVE WHEN WE INITIATED TURN; APCHING ACFT BECAME AWARE OF US AND INITIATED A R TURN INTO OUR PATH AS WELL AS A DSCNT; POSSIBLY THINKING THIS WAS THE SAFEST ACTION TO TAKE. I SAY THIS BECAUSE I BELIEVE I HEARD CTLR SAYING HE HAD STARTED TO DSND. ANOTHER REFLECTION: TCASII WAS VERY LOUD AND BTWN TCASII AND CTLR ISSUING INSTRUCTIONS; I BELIEVE IT BECAME ALMOST A MATTER OF PANDEMONIA (AS IT REALLY WAS). THE CTLR WAS EXCELLENT; BUT I THINK WHILE HE WAS REACTING CORRECTLY TO WHAT HIS SOFTWARE WAS TELLING HIM; AT OUR INITIAL HIGH RATE OF CLB A TURN AND A CONTINUED CLB MAY HAVE AVOIDED A VERY CLOSE ENCOUNTER. ALSO; HAVING COME UP FROM GA; OTHER PLT PROBABLY DID WHAT AIM SAYS WHEN FACING HEAD ON TFC; ROLLING R AND PROBABLY DIVING TO AVOID HIT.