PA34 PRACTICE APCH TO RAP EXPERIENCED NMAC ON SHORT FINAL.

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2 PLTS; NO PAX ON BOARD. PLT #1 PIC AND SAFETY PLT. PLT #2 FLYING PRACTICE ILS RWY 32 TO RAPID CITY REGIONAL ARPT USING VISION LIMITING DEVICE. THIS WAS THE THIRD PRACTICE APCH DURING THIS SESSION. WE WERE UNDER VFR FLT RULES. HOWEVER; WE WRE TALKING TO ELLSWORTH APCH/DEP CTL AND TO THE RAPID CITY TWR. ELLSWORTH VECTORED US FOR THE ILS RWY 32 APCH; CLRED US FOR THE APCH AND HANDED US OFF TO THE TWR. WE CONTACTED THE TWR. THEY ADVISED US TO 'CALL RANCH OM INBOUND.' WE CALLED 'RANCH' AND TWR CLRED US FOR THE OPTION TO RWY 32. AS WE WERE NEARING THE END OF THE APCH; ANOTHER ACFT CONTACTED THE TWR AND XMITTED SOMETHING TO THE EFFECT 'WHERE DO I GO; THERE IS ANOTHER ACFT ON SHORT FINAL?' AT THE SAME TIME; I (PLT #1) SAW THE ACFT AND IMMEDIATELY TOLD THE PLT (#2) FLYING 'MY AIRPLANE.' I PWRED UP AND STARTED A CLBING L TURN TO AVOID ANY FURTHER POTENTIAL CONFLICT. WE PASSED 150-200 FT DIRECTLY OVER THE TOP OF THE OTHER ACFT THAT WAS TURNING A SHORT FINAL FOR RWY 32. WE CONTACTED THE TWR AND SAID 'WHERE DID THAT GUY COME FROM?' NO RESPONSE FROM THE TWR. WE THEN CALLED A MISSED APCH AND TWR HANDED US OFF TO ELLSWORTH WITH NO OTHER COMMENT. WE DO NOT KNOW WHERE THE OTHER ACFT WENT. I ASSUME HE LANDED. WE DO NOT REMEMBER ANY TWR COMS WITH THE OTHER ACFT. WE DID NOT RECEIVE ANY TA FROM THE TWR ABOUT THIS ACFT.