RPTED SIT IN WHICH A CPR LR31 AT 7000 FT IS GIVEN CLB CLRNC AT ABOUT THE SAME TIME THE FLC IS RESPONDING TO A TCASII TA RA. NO INDICATION OF LTSS.

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DEPARTED TEB IFR FLT PLAN ON THE TEB 4 DEP; WHITE INTXN. FO WAS FLYING THE ACFT. HDOF TO NEW YORK DEP FROM TWR; GIVEN VECTORS AND ALT ASSIGNMENTS. DEP CTLR #2 CLRED US TO 17000 FT. DURING THE CLB; HE ASKED US TO STOP AT 7000 FT. WE WERE AT 7300 FT BY THAT TIME SO I READ BACK AND WE DSNDED BACK TO 7000 FT. HE THEN HAD US CONTACT THE THIRD DEP CTLR. DURING THE HDOF WE BEGAN TO GET A TA OFF OUR TCASII; THEN; AS WE BEGAN TO RECEIVE AN RA; I COULD HEAR THE CTLR TRY AND GIVE CLBING INSTRUCTIONS BUT HE WAS BLOCKED BY ANOTHER ACFT XMISSION. WE THEN BEGAN TO TAKE OUR OWN ACTION RESPONDING TO OUR TCASII. I COULD TELL HE HAD TO BE TALKING TO OUR ACFT SO I SAID BLOCKED AND ASKED HIM TO REPEAT. AT THAT TIME HE GAVE DIRECT CLB; CLB INSTRUCTIONS PLUS A 180 DEG HDG TO OUR ACFT. I ACKNOWLEDGED THE INSTRUCTIONS AND COULD SEE MY FO WAS NOT RESPONDING QUICK ENOUGH WITH THE ACFT SO I TOOK OVER; IMMEDIATELY BEGAN A 6000 FPM CLB WITH A 180 DEG HDG. WE AVOIDED THE TFC AS I COULD SEE HIM AT MY 2 O'CLOCK POS LOW. I CALLED ATC THAT EVENING AND TALKED TO THE ATC MGR. WE TALKED ABOUT WHAT HAPPENED; HE ACKNOWLEDGED THAT IT WAS A MISTAKE ON THEIR PART; BY NOT BRIEFING THE THIRD DEP CTLR THAT WE WERE TO STAY AT 7000 FT. I DID CHK IN WITH THE THIRD CTLR AND SAID WE WERE AT 7000 FT. WE WERE NOT CLBING OR DSNDING OUR ACFT BEFORE THE OCCURRENCE HAPPENED.