ACR X TCASII RA DSND HAD LTSS FROM WDB DSCNT THROUGH OCCUPIED ALT. SYS ERROR. EVASIVE ACTION TAKEN.

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ACR X ON A VECTOR FROM ZNY OF 060 DEGS R OFF OF AIRWAY. RECEIVED INSTRUCTIONS TO TURN L TO INTERCEPT J-174 (ORIGINAL ROUTING). JUST AS THE CTLR SAID TURN L TO ...; THE TCASII ISSUED AN RA. THERE WAS NO TA. I LOOKED DOWN ON THE RADAR SCOPE AND NOTED TFC AT 10 DEGS L; 6 NM; 1100 FT HIGH AND IN A DSCNT. I ROLLED OUT OF THE L TURN; UNCOUPLED THE AUTOPLT AND BEGAN A DSCNT. THE VVI INDICATED A 1500-2000 FPM GREEN ARC. THE TFC WAS NOT VISIBLE DUE TO HIGH CIRRUS CLOUDS AT FL350. AT 2 NM THE TFC WAS NOTED AT 800 FT HIGH AND LEVELED OFF AS WE HAD DSNDED TO FL342. TCASII ISSUED A MONITOR VERT SPD COMMAND JUST AS I LOOKED OUT THE L WIND SCREEN AND NOTED A WDB SEEMINGLY LEVEL AND WITHIN 1/4-1/2 NM. ATC WAS NOTIFIED OF THE CLOSE ENCOUNTER AS WE LEVELED OFF AT FL342. WE THEN CLBED TO FL350 AND INITIATED THE L TURN TO INTERCEPT J-174 N OF SWL. SUPPLEMENTAL INFO FROM ACN 238204: ACR X PASSING 34300 FT WE SAW A WDB PASS OFF OUR L WING WITHIN 1/2 MI AND STILL DSNDING. WE LEVELED OFF AT 34200 FT AFTER TCASII NOTIFIED CLRED OF CONFLICT. WE NOTIFIED ZNY OF THE EVASIVE ACTION AND WAS CLRED BACK UP TO FL350. WE BELIEVE IF WE HADN'T HAD TCASII ON BOARD THINGS WOULD HAVE BEEN MUCH CLOSER THAN THEY OCCURRED. ZNY THANKED US FOR OUR HELP AND CONFIRMED THE EVENT.