ZJX CTLR EXPERIENCED OPERROR AT APPROX FL330 WHEN INCORRECT ASSIGNED ALT READ BACK WAS INCORRECT AND MISSED DURING RELIEF BRIEFING.

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AT APPROX XA10 LCL TIME I WAS WORKING THE R75 SECTOR. TFC VOLUME WAS SLOWING DIMINISHING TO MODERATE FROM VERY HVY. THE R57 SECTOR CALLED ME AND REQUESTED HIGHER ON ACFT #1; WHO THEY HAD TURNED 15 DEGS R FOR THE CLB. I APPROVED FL270. AT THAT TIME; ANOTHER CTLR PLUGGED INTO R75 TO RELIEVE ME. I BEGAN A RECORDED POS RELIEF BRIEFING. ABOUT HALFWAY THROUGH THE BRIEFING ACFT #1 CHKED ONTO MY FREQ CLBING TO FL270 STILL ON A HDG. I TOLD THE ACFT TO FLY HDG 355 DEGS; CLB AND MAINTAIN FL330. APPARENTLY THE PLT READ BACK THE CORRECT HDG; BUT STATED FL390 INSTEAD OF FL330. I DID NOT CATCH THIS MISTAKE; NOR DID THE RELIEVING CTLR. I FINISHED THE BRIEFING AND LEFT THE SECTOR. A FEW MINS LATER ACFT #1 CLBED THROUGH FL330 AND RPTED THAT THEY WERE RESPONDING TO A TCAS RA AND WERE DSNDING BACK DOWN TO FL330. THE CTLR STATED THAT THEY WERE ASSIGNED FL330 AND TO MAINTAIN FL330. ACFT #2 WAS OPPOSITE DIRECTION AT FL340 AND CLBED IN RESPONSE TO TCAS RA WHEN ACFT #1 LEFT FL330. 1000 FT OF VERT SEPARATION WAS MAINTAINED. SEPARATION WAS NEVER LOST. CONTRIBUTING FACTOR WAS THAT I WAS CONDUCTING BRIEFING AT THE TIME AND THAT THE PLT WHO READ BACK FL390 HAD A STRONG ACCENT (FOREIGN ACR). SUPPLEMENTAL INFO FROM ACN 731575: 1) I WAS FLYING OVER GEORGIA SPEAKING TO ZJX. MY FO WAS ON THE OTHER RADIO TALKING TO FLT WATCH. MY FLT LEVEL WAS 36000 FT. ZJX ASKED ME IF I COULD ACCEPT FL380 FOR TFC. I LOOKED AT MY WT ON THE FMS AND SAID YES; BUT IT WOULD BE A SLOW CLB. HE SAID I WOULD HAVE TO TURN 30 DEGS FOR A VECTOR. I SAID I COULD DO 500 FPM INSTEAD OF A VECTOR. HE SAID OK (MAYBE YES) OR WORDS TO THAT EFFECT. SO I STARTED TO CLB TO FL380. AT ABOUT 36300-36400 I GOT A YELLOW TFC WARNING ON TCAS AT ABOUT 10 MI TO MY L (11 O'CLOCK POS). I WAS HDG S FROM UGN-O7FA. 2) OTHER ACFT TOLD ATC HE WAS GETTING A WARNING. HE TURNED 30 DEGS AND ATC TOLD ME TO TURN 30 DEGS R; TO WHICH I RESPONDED AND WAS TOLD TO GO BACK DOWN TO FL360. HE MAY HAVE ASKED MY ALT BEFORE THAT I AM NOT SURE. THE OTHER ACFT LATER ASKED ZJX FOR A PHONE NUMBER. 3) I BELIEVED AT THE TIME I WAS CLRED TO FL380 WHEN I SAID I COULD CLB AT 500 FPM WOULD THAT BE OK AND THE CTLR SAID YES. 4) I THOUGHT THAT THERE WAS TFC AT MY ALT AND THE CTLR WANTED ME HIGHER OR DEVIATING TO AVOID A CONFLICT.