SMA X HAD NMAC WITH Y IN TFC PATTERN. SEE AND AVOID CONCEPT. EVASIVE ACTION TAKEN.

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SMA X WAS TOLD TO ENTER DOWNWIND AND FOLLOW TFC TURNING XWIND. WHEN 2 MI S OF THE ARPT; WE ACQUIRED TFC AND ADVISED TWR. ANOTHER ACFT IMMEDIATELY CALLED TWR; SO WE DID NOT RECEIVE A RESPONSE. CONFIDENT OF OUR TFC; WE FOLLOWED THEM ON A L DOWNWIND FOR RWY 23. IT BECAME CLR THAT THEY WERE EITHER EXTENDING THEIR DOWNWIND OR DEPARTING THE PATTERN TO THE E. I WAS ABOUT TO ASK ATC WHEN THE ACFT BEHIND ME ASKED IF I WAS DEPARTING. THE TWR RESPONDED 'CESSNA TFC WAS DEPARTING TO THE E. DO NOT FOLLOW.' I WAS FOLLOWING THE CESSNA TFC; SO AFTER FIRST ACFT RESPONDED; I ALSO ACKNOWLEDGED THE MESSAGE. I THEN LOOKED FOR TFC ON FINAL AND TURNED L BASE. I WAS INSTRUCTING MY STUDENT ON HOW TO DO HIS FIRST LNDG WHEN THE CTLR WAS REPLACED BY A NEW VOICE WHO ADVISED THE ACFT BEHIND ME TO DO A 360 AND RE-ENTER BASE. THE ACFT RESPONDED THAT HE WAS NOT ON BASE. ATC THEN ASKED 'ACFT ON R BASE; SAY IDENT.' THERE WAS NO REPLY. HE REPEATED HIS REQUEST 2 MORE TIMES. I THEN IDENTED MYSELF AS BEING ON L BASE. AT ABOUT THE SAME TIME; MY STUDENT NOTICED ANOTHER ACFT Y AT OUR ALT COMING AT US ON FINAL. MY FIRST INSTINCT WAS TO TURN; WHICH WAS WRONG BECAUSE IT BROUGHT US CLOSER. I IMMEDIATELY REALIZED MY MISTAKE AND ADDED PWR AND CLBED TO PATTERN ALT. ATC INSTRUCTED ME TO THEN ENTER A R DOWNWIND. HALF OF THE TWR CTLRS ARE 'NEW HIRES.' THEY WERE HIRED LAST JUNE AND HAVE NEVER HAD TO DEAL WITH THE UNIVERSITY TFC. BECAUSE I NOT ONLY HAVE TO INSTRUCT MY STUDENT BUT WATCH FOR ERRORS MADE BY ATC; I AM NOT TRYING TO BLAME THE WHOLE EVENT ON THE CTLR. I REALIZE TFC AVOIDANCE IS MY RESPONSIBILITY AND IF I HAD NOT BEEN SO SURE OF THE TFC SIT AND SO INTENT ON INSTRUCTING MY STUDENT; I MAY HAVE SEEN THE OTHER TFC.