C172 PLTS HAVE AN NMAC DURING APCH TO PRC.

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WE WERE INBOUND TO THE ARPT ON AN IFR FLT PLAN. WE WERE CLRED FOR THE VOR RWY 12 APCH INTO PRC. PRC TWR WAS EXTREMELY BUSY AND WE COULD NOT GET OUR CALL IN ON THE FREQ UNTIL REACHING THE INTERMEDIATE FIX OF THE APCH. TWR HANDED US OVER TO THEIR 2ND FREQ; WHICH WAS ALSO VERY BUSY. WE COULD NOT GET OUR CALL IN UNTIL THE FINAL APCH FIX (DRK VOR). WE TOLD TWR WE WOULD CANCEL IFR; AND WE WERE CLRED FOR THE VOR RWY 12 PRACTICE APCH; WITH AN ALT RESTR TO MAINTAIN AT OR ABOVE 6500 FT MSL. UPON REACHING THE MISSED APCH POINT; I NOTICED ON OUR ADS-B DISPLAY THERE WAS A TARGET HEADED TOWARDS US; INDICATED AS BEING +300 FT ABOVE OUR ALT. THE ACFT WAS COMPANY TFC; ALSO WITH ADS-B. HAD WE EXECUTED THE PUBLISHED MISSED APCH PROC AT THIS TIME; IT WOULD HAVE FLOWN US HEAD-ON INTO THE TFC. I MADE VISUAL CONTACT JUST AS MY STUDENT WAS ABOUT TO EXECUTE THE MISSED APCH PROC; A CLBING L TURN (RIGHT INTO TFC). THE TFC APPEARED TO BE HIGHER; SO I TOOK CTLS TO PREVENT US FROM TURNING/CLBING INTO HIM. (THIS WAS ALL IN THE TIME SPAN OF 5 SECONDS OR LESS.) THE TFC PASSED DIRECTLY OVERHEAD; ESTIMATED DISTANCE OF 100 FT VERT. TWR DID NOT RPT THE TFC TO US UNTIL HE WAS PASSING OVERHEAD AND THE CTLR EVEN APOLOGIZED FOR NOT RPTING IT SOONER. AFTER THE FLT; I DISCUSSED THE INCIDENT WITH THE PIC OF THE 2ND ACFT. HE SAID THEY WERE ON THE OTHER TWR FREQ (125.30; WE HAD BEEN ON 128.75). HE ALSO SAID HIS CTLR HAD NOT RPTED THE TFC TO HIM AT ALL. THEY WERE ALSO RESTR AT OR ABOVE 6500 FT MSL.