NMAC BTWN A GULFSTREAM IV; G4; DSNDING TO FINAL ILS APCH ALT AND ANOTHER VFR ACFT CLBING THROUGH THE ALT TO WHICH THEY WERE DSNDING. THE G4 MADE A HIGH RATE OF CLB TO AVOID A COLLISION.

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WE WERE FLYING FROM STW DIRECT TO WANES (INTXN) FOR THE VOR/DME TXWY A APCH TO TEB. WE WERE WORKING NY APCH ON 127.3 AND THE CTLR WAS VERY BUSY. WE WERE DSNDING TO 3000 FT -- APCH CALLED OUT ACR TFC ABOVE US; 11 O'CLOCK POS -- WE HAD THE TFC VISUALLY. HOWEVER; OUT OF 3500 FT WE SAW A TCASII TARGET AT 12 O'CLOCK POS; TCASII SHOWED -500 FT AND ABOUT 1 MI. I WAS SLOWING THE DSCNT TO 1000 FPM FOR LEVELOFF; TCASII CALLED TFC; TFC. OTHER CAPT IN R SEAT (I WAS L SEAT) COULD NOT FIND THE VFR TFC; NEITHER COULD I. IN LESS THAN 3 SECONDS TCASII CALLED CLB; CLB; CLB -- TCASII DISPLAY SHOWED THE TFC AT 12 O'CLOCK POS AND -100 FT. THE BOGY SYMBOL WAS AT THE NOSE OF OUR 'A' SYMBOL ON TCASII. I PUSHED PWR TO 'GAR' AND PITCHED TO 10 DEGS. WE NEVER DID SEE THE VFR TFC. THE G4 WAS LIGHT; MINIMUM FUEL AND 2 PAX. THE AIRPLANE CLBED VERY QUICKLY; I IMMEDIATELY STARTED REDUCING PWR AND PUSHED THE NOSE TO LEVEL. I SAW 4200 FT ON ALTIMETER. R SEAT PLT WAS ON RADIO TELLING CTLR OF OUR RA. I HAD THE NOSE DOWN; PWR OFF; AND DSNDING BACK TO 3000 FT. CTLR WAS VERY CONCERNED WE CLBED HIGHER THAN 500 FT TO MISS THE TFC. HE AGAIN CALLED THE ACR WHICH WE HAD IN SIGHT AND WAS NOT A FACTOR FOR US. WE WERE TRYING TO AVOID THE VFR AIRPLANE WHICH BECAME A PROB FOR US IN LESS THAN 10 SECONDS AFTER WE FIRST SAW THE TCASII TARGET. THE PROB WAS HOW QUICK THE VFR TARGET BECAME A THREAT; THE FACT HE WAS 12 O'CLOCK POS; CLOSE AND 100 FT BELOW OUR ALT. WE STILL HAD 200 FT TO DSND TO OUR ASSIGNED 3000 FT. SEEMED LIKE A COLLISION TO ME; AND TO TCASII; BECAUSE IT WENT FROM 'TFC' TO 'CLB; CLB; CLB' IN LESS THAN 3 SECONDS. PROB WAS THE G4 WAS LIGHT AND CLBED HIGHER AND FASTER THAN I HAD PLANNED. STILL; WE WERE BACK AT 3000 FT IN ABOUT A MIN. THE AREA WAS VERY BUSY WITH VFR AND IFR TFC BUT THE REST OF THE APCH AND LNDG WAS UNEVENTFUL. THE CTLR WAS BUSY BUT WAS DOING A GOOD JOB OF CTLING. I AM STILL NOT SURE HE KNEW HOW VERY CLOSE THE VFR TFC WAS TO OUR AIRPLANE.