A MOONEY PVT PLT HAS A PROB WITH 2 CTLRS IN AN ESCAPE ACTION AFTER HE GETS TRAPPED BTWN 2 CPR ACFT ON FINAL APCH TO RWY 16 AT HPN; NY.

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APCHING HPN FROM THE E. WE HAD ATIS INFO. WINDS 160 DEGS AT 5 MI; VISIBILITY 10 MI; CEILING 12000 FT. CHKED IN WITH APCH CTL ON 126.4 FOR SEQUENCING. TOLD TO FLY TO FIELD FOR L DOWNWIND RWY 16. PASSED OVER TO TWR. TWR ASKED IF RWY 11 WAS ACCEPTABLE. I DECLINED; ASKED FOR RWY 16. TWR ASKED FOR L 360 DEG TURN FOR SPACING BEFORE CONTINUING DOWNWIND. I COMPLIED. TWR ADVISED I WAS #3 FOR RWY 16; FOLLOWING GULFSTREAM TFC ON ILS. I ASKED IF TWR WOULD CALL MY BASE. TWR SAID 'NO; YOU MUST TELL ME WHEN YOU SEE YOUR TFC.' ON DOWNWIND; ABOUT 1/R MI PAST ARR END OF RWY 16; I SAW A JET ON THE ILS; OFF MY L WINGTIP. I SAID TO TWR; 'TWR; N-NUMBER SEES ACFT OFF MY L WING; IS THAT THE TFC I AM FOLLOWING?' TWR SAID; 'AFFIRMATIVE N-NUMBER.' I AID TO TWR; 'N-NUMBER TURNING BASE NOW TO FOLLOW TFC.' TWR DID NOT RESPOND. I TURNED BASE AND IMMEDIATELY SAW ANOTHER ACFT BEARING DOWN ON ME. I SAID TO TWR; 'N-NUMBER SEES TFC INBOUND RIGHT AT US; BREAKING OFF.' I IMMEDIATELY CONFIGURED FOR FULL PWR CLB ALONG BASE LEG. ANOTHER VOICE (SECOND CTLR) ON TWR FREQ SAID; 'N-NUMBER I NEED YOU TO PROCEED ONTO FINAL.' I REPEATED; 'TFC OFF MY R SIDE; CONTINUING ON BASE.' SECOND CTLR THEN SAID; 'OK N-NUMBER; TURN R TO 340 DEGS FOR AN EXTENDED NOW R DOWNWIND FOR RWY 16.' I RECONFIGURED FOR DOWNWIND. FIRST CTLR THEN SAID; 'N-NUMBER TURN TO FINAL AT ONCE AND STRAIGHT FOR THE NUMBERS; I NEED YOU TO HURRY BECAUSE YOU ARE IN THE BOX.' I STARTED TO COMPLY; ONLY TO SEE ANOTHER ACFT DEAD AHEAD. I ADVISED TWR; 'TFC AT 12 O'CLOCK POS; TURNING L (AWAY FROM FIELD) TO AVOID.' SECOND CTLR SAID; 'N-NUMBER; TURN TO 270 DEGS IMMEDIATELY.' I COMPLIED. AFTER FOLLOWING 270 DEG HDG; AND CALMING DOWN; I SAID TO TWR; 'N-NUMBER WILL PROCEED TOWARD TAPPAN ZEE BRIDGE AND RE-ENTER ARPT AREA FOR RWY 11.' CTLR AGREED. WE LANDED WITH NO FURTHER INCIDENT. I MISIDENTED TFC ON FINAL AS THE GULFSTREAM I WAS TO BE FOLLOWING. THIS WAS MY ERROR. TURNS OUT THE TFC WAS ANOTHER TURBINE BUSINESS CLASS JET. HAD THE CTLR AGREED TO CALL MY BASE WHEN SEQUENCING ME INTO COMPLEX STREAM OF INBOUND TFC ON ILS; THE WHOLE INCIDENT WOULD HAVE BEEN AVOIDED. I; LIKE MOST GA PLTS; HAVE TROUBLE POSITIVELY TELLING THE DIFFERENCE BTWN A G-V AND LEAR AND CITATION AT 1 OR 2 MI -- ESPECIALLY WHEN BUSY PREPARING FOR LNDG. I NEEDED HELP. FIRST CTLR WAS NOT PAYING ATTN TO MY SIT. SECOND CTLR WAS NOT COORDINATING WITH FIRST CTLR. BETTER COORD COULD HAVE AVOIDED THE SECOND TARGET. AND -- WHERE IS THE BOX? HOW SHOULD I KNOW WHAT IT MEANS OR WHAT TO DO ABOUT IT? ALL THIS LANGUAGE DID WAS ADD TENSION TO AN ALREADY TENSE SIT. THESE HPN CTLRS WORK HARD; AND THIS WAS A BUSY TIME. HOWEVER; THIS WAS A CLOSE CALL IN MY VIEW. IT COULD HAVE BEEN AVOIDED BY BETTER CTLR/PLT COM AND COORD.