WDB FLC AT CRUISE RECEIVES TCASII TA. TURNED TO ASSURE SIGHTING AND ACQUIRED VISUALLY. MAINTAINING SEPARATION BUT RECEIVED RA; THEN QUICKLY CLR OF CONFLICT.

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DURING NORMAL CRUISE AT FL370 SBOUND ON A401 (AIRWAY NAME QUESTIONABLE SINCE NOT IN POSSESSION OF AIRWAYS CHART AT THIS WRITING). APCHING RESAL INTXN. VMC CONDITIONS. AURAL TA ON TCASII -- 'TFC; TFC.' TFC NOTED ON TCASII SCREEN AT 12 O'CLOCK POS; APPROX 20 MI AND LEVEL WITH OUR ACFT. WE STRAINED TO VISUALLY SEE THE OTHER ACFT WHILE THE COPLT (PF THIS LEG) ASSUMED CTL OF THE AFDS PANEL AND DIRECTED A 30 DEG HDG CHANGE TO THE R. VISUALLY SIGHTED OTHER ACFT NEAR THE 11:30-12 O'CLOCK POS; PROCEEDING NBOUND; IN THE OPPOSITE DIRECTION. I CONTINUED TO MONITOR OTHER ACFT TO ENSURE IT'S IMAGE TRAVERSING ACROSS OUR COCKPIT WINDSCREEN FROM R TO L; AS THE OTHER ACFT APCHED OUR 10 O'CLOCK POS; AND TO OUR SATISFACTION; PASSING CLR OF OUR ACFT THE TCASII CHANGED TO AN RA 'DSND; DSND' WITH A RED SYMBOL ON THE TCASII SCREEN. THIS WAS FOLLOWED BY A TCASII AURAL 'CLR OF CONFLICT' STATEMENT AS THE OTHER ACFT PASSED CLR. I IMMEDIATELY CALLED BRAZILIA CTR ON VHF AND ADVISED OF THE CONFLICT. THE CONVERSATION NOW CHANGED FROM ENGLISH TO PORTUGUESE AS THE CTLR DISCUSSED WITH THE OTHER ACFT. SHORTLY THE CTLR CALLED AND PROVIDED THAT THE OTHER ACFT WAS FLT XX WHO WAS 'CLBING TO FL390.' A STATEMENT WAS MADE SUGGESTING THAT SOMEONE 'FORGOT...' THE XX ACFT BUT IT WAS UNCLR WHETHER IT WAS ATC OR THE OTHER ACFT THAT HAD 'FORGOT.' (IT SHOULD BE NOTED THAT THE OTHER ACFT HAD BEEN OPERATING IN AN ATC RADAR ENVIRONMENT UNTIL A SHORT WHILE BEFORE; SINCE ATC RADAR DOES NOT EXTEND AS FAR N AS TESAL INTXN). AFTER A HF PHONE PATCH WITH OUR COMPANY OPS TO RPT THE INCIDENT; OUR FLT CONTINUED ON TO IT'S DEST AT FL370 AS ASSIGNED. THE APPROPRIATE WRITTEN RPTS WERE FILED BY THE CAPT UPON FLT COMPLETION.