NMAC BTWN SF34L AND B737. SF34L CREW FOLLOW TCASII RA. CTLR ISSUES A SERIES OF RAPID ALT AND HEADING CHANGES TO BOTH ACFT.

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ON FLT XXX FROM DCA TO BOS WE WERE INSTRUCTED TO DSND FROM 9000 FT TO 7000 FT BY PROVIDENCE APCH. AS WE BEGAN OUR DSCNT WE WERE TO LOOK FOR AN ACR ZZZ B737 AT OUR 2 O'CLOCK POS CLBING FROM 5000 FT TO 8000 FT. WE ADVISED THAT WE DIDN'T HAVE THE TFC AND WERE IMC AT THE TIME. WHILE DSNDING THROUGH 7700 FT WE BROKE OUT OF THE CLOUDS. TCASII SHOWED OUR TFC AT 2 O'CLOCK POS. IMMEDIATELY ATC TOLD US TO STOP OUR DSCNT AT 9000 FT. WE INFORMED HIM WE WERE AT 7700 FT. CTLR TOLD ACR ZZZ TO CLB; NO DELAY; TO 15000 FT. AT THAT TIME OUR TCASII SHOWED THEM AT 7100 FT. PROVIDENCE TOLD US TO CLB TO 9000 FT AND TURN L TO 360 DEGS; THEN 300 DEGS; FINALLY 270 DEGS AND TOLD US TO MAINTAIN 7000 FT. WE HAD THE ACFT IN SIGHT AND DSNDED TO 6700 FT DURING THE TURN TO AVOID THEM; THEN IMMEDIATELY CLBED BACK TO 7000 FT. DURING THE TURN OUR TCASII TOLD US TO CLB AND BANK L. HOWEVER; WE HAD THE ACFT IN SIGHT (TCASII SHOWED THEM AT 7100 FT); SO WE BANKED 30 DEGS L INITIALLY TO 360 DEGS AND DSNDED TO 6700 FT. WE ADVISED ATC OF THIS AND THEY TOLD US TO KEEP OUR L TURN COMING AROUND TO DIRECT BOS.