C172 INSTRUCTOR REPORTS CONFLICT WITH ANOTHER C172 IN THE PATTERN AFTER LOSING SIGHT OF TRAFFIC AND TURNING CROSSWIND EARLY.

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IT WAS A VERY BUSY DAY AT ZZZ. THERE WERE AT LEAST 4 PEOPLE IN THE PATTERN AND SEVERAL MORE SHOOTING APCHS. I WAS FLYING WITH A VERY LOW TIME PVT STUDENT AND WE HAD JUST TAKEN OFF. INTENDING TO STAY IN THE PATTERN; I WAS WATCHING THE TFC AHEAD OF US WHO WAS ALSO ON AN UPWIND; AND WAS ALSO STAYING IN THE PATTERN. I NOTICED A SLIGHT BUFFETING OF THE ACFT; AND LOOKED DOWN TO DISCOVER THAT MY STUDENT HAD THE PLANE VERY NEAR THE STALL SPD. I MOMENTARILY TOOK CTL OF THE ACFT AND REGAINED A STABLE CLBOUT. DURING THIS TIME I LOST SIGHT OF THE TFC AHEAD OF ME. AS I APCHED PATTERN ALT I SAW WHAT I THOUGHT WAS MY TFC TO FOLLOW PASS BEHIND ME ON THE DOWNWIND. IN REALITY; IT WAS THE PLANE IN FRONT OF MY TFC. AND BY TURNING TO FOLLOW HIM; MY XWIND TOOK ME DIRECTLY IN FRONT OF THE PLANE AHEAD OF ME WHO HAD TURNED DOWNWIND. THE END RESULT WAS ME XING HIS PATH AND TURNING PARALLEL TO HIS DOWNWIND AND WE WERE FLYING A FORMATION PATTERN WITH THE OTHER TRAINER LESS THAN 200 FT OFF MY L WING. I DEPARTED THE TFC PATTERN TO THE N AND COLLECTED MY CONSIDERABLY SHAKEN SELF AND RETURNED TO LAND UNEVENTFULLY AT THE FIELD. IN THE FUTURE; I WILL TRY MUCH MORE DILIGENTLY TO MAINTAIN MY SITUATIONAL AWARENESS AND KEEP TRACK OF EXACTLY HOW MANY PEOPLE ARE IN THE PATTERN AND WHERE THEY ARE IN RELATION TO ME. I WILL ALSO GO TO A SMALLER; LESS CROWDED ARPT FOR PATTERN WORK DURING THE PEAK TRAINING HRS SO AS TO AVOID THE CLUSTER WHICH RESULTED.