ACR X TCASII RA HAD LTSS FROM ACFT Y. SYS ERROR. EVASIVE ACTION TAKEN.

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APPROX 40 MI SW OF LGA ARPT; CLBING THROUGH 13000 FT TO 17000 FT; WE EXPERIENCED TCASII ADVISORY OF ACFT AT 12 O'CLOCK DSNDING THROUGH 15300 FT. RECEIVED SECOND TCASII ADVISORY; FOLLOWED BY CTR CTLR INSTRUCTIONS TO 'EXPEDITE CLB THROUGH 15000 FT.' IMMEDIATELY INITIATED RAPID CLB. SHORTLY THEREAFTER; RECEIVED TCASII RA TO 'DSND.' DUE TO OUR VERY HIGH RATE OF CLB; WE CONTINUED WITH THE CTLR'S RESOLUTION. ACFT WAS VISUALLY SIGHTED AT 1 O'CLOCK AND LESS THAN 1 MI; PASSING OUR ALT. SUPPLEMENTAL INFO FROM ACN 266648: THE ZNY CTLR INSTRUCTED US TO EXPEDITE OUR CLB TO 17000 FT AND WE INCREASED OUR CLB RATE TO 3500 FPM. AFTER WE INCREASED THE RATE OF CLB TO 3500 FPM; WE RECEIVED A TCASII RA TO DSND AT 2500 FPM. AT THIS TIME WE DETERMINED THAT THE TFC WAS ROUGHLY 14000 FT AND AT OUR 12 O'CLOCK POS. WE NEXT SPOTTED THE ACFT VISUALLY AT 12 O'CLOCK AND DECIDED TO CONTINUE OUR CLB. WE ALSO DECIDED TO INITIATE A R TURN TO AVOID THE TFC. THE TCASII WAS AN INVALUABLE AID BUT THE RA CAME AFTER WE WERE COMMITTED TO CLB. THE OTHER ACFT WAS DSNDING.