A320 AND B727 CONFLICT WHEN B727 IS VECTORED AGAIN TO THE RWY 12 FINAL APCH COURSE.

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ON 300 DEG VECTOR ABEAM WASHY INTXN FOR ARR TO ILS RWY 12L MSP. WE WERE DSNDING TO 5000 FT WHEN ATC ISSUED A REQUEST TO TURN TO 360 DEG HDG; CLB TO 6000 FT. SIMULTANEOUSLY; WE RECEIVED A TCASII RESOLUTION TO 'CLB; INCREASE CLB.' CONFLICT WAS OF SHORT DURATION AND WE NOTICED ON OUR NAV DISPLAYS THE OTHER ACFT WAS 300 FT BELOW US. ALSO; I SAW THE ACFT AT THIS TIME ON AN APPARENT MISSED APCH TO RWY 12L. THE ACFT WAS IN A CLBING L TURN AT APPROX 5000 FT. WE COMPLIED WITH THE RA BY DISCONNECTING THE AUTOPLT AND MAKING A CLBING TURN TO 360 DEGS AND 6000 FT -- AS REQUESTED BY ATC (MSP APCH CTL). RESOLUTION WAS SATISFIED AND WE WERE THEN VECTORED FOR APCH TO RWY 12L ILS MSP. SUPPLEMENTAL INFO FROM ACN 497884: FLT WAS BEING RADAR VECTORED FOR THE ILS TO RWY 12R APCH. ATC WAS GIVING US S-TURNS WITHIN 15 MI FROM ARPT. APPROX 4 S-TURNS WERE GIVEN TO US AND WE WERE IN IMC. WE ASKED TO BE BROUGHT BACK AROUND AND START OVER; SO ATC CLBED US UP TO 5000 FT; THEN HANDED US OFF TO ANOTHER CTLR. UPON REACHING 5000 FT; THE TCASII WAS GIVING US AN RA AND THEN WE WERE CLRED BACK TO 4000 FT. THE S-TURNS WERE TOO MUCH FOR ATC TO ASK OF US IN IMC. THE SAFEST THING TO DO IS TO BE BROUGHT BACK AROUND FOR THE APCH.