PA28 LNDG MLB IS REQUIRED TO EXECUTE A 360 DEG TURN TO AVOID TFC.

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MY INSTRUCTOR AND I WERE FLYING BACK TO OUR HOME BASE TO LAND. WE CALLED ATC ABOUT 5 NM TO THE S OF THE ARPT. WE CALLED IN AS XABAXX AND THE CTLR ANSWERED US AS XABAXX. HE INSTRUCTED US TO FLY L BASE RWY 27L AND RPT OVER THE CAUSWAY. PRIOR TO REACHING CAUSWAY; THE CTLR WAS ATTEMPTING TO CONTACT ACFT XAEEXX. THE CTLR ATTEMPTED TO GET IN TOUCH WITH XAEEXX 3 TIMES WITH NO RESPONSE. AT THE SAME TIME AN ARROW; XFFEXX WAS ON A L DOWNWIND FOR RWY 27L. WHEN WE CALLED THE CTLR TO RPT OUR POS OVER THE RWY; HE SAID THAT HE WAS TRYING TO CONTACT US AND THAT WE SAID WE WERE XAEEXX; AND WE VERIFIED THAT WE WERE XAEEXX. AT THE SAME TIME; THE ARROW ON DOWNWIND WAS ON A COLLISION COURSE WITH OUR FLT PATH. WE DECIDED THAT IT WOULD BE BEST IF WE DID A 360 DEG TURN TO AVOID THE ACFT AND HAVE A SAFE SEPARATION. WHEN WE STARTED TO DO OUR 360 DEGS; THE CTLR SAID THAT WE COULD NOT DO THAT; THAT WE MUST FLY NBOUND AND FOLLOW THE TFC. WE CAME VERY CLOSE AND WE WERE PLACED IN A VERY UNCOMFORTABLE POS. WHEN WE WERE BACK ON THE GND; WE VERIFIED WITH OTHER PLTS IN THE PATTERN; WHO ASSURED US THAT WE MADE CONTACT AS XABAXX; NOT XAEEXX.