MU300 CREW HAD AN NMAC WITH A B777-200. THE MU300 CREW DID NOT ENCOUNTER FORECAST OUTSIDE AIR TEMPS; OR FORECAST TAILWINDS. THEY WERE NOT ABLE TO CONTACT THE CZQX NON RADAR CTLR. THEY UNILATERALLY CLBED TO 36500 FT.

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WHILE ENRTE TO SONDRESTROM; DUE TO LACK OF RADIO COM ABILITY AND UNFORECAST 1SO TEMPS AS WELL AS RADICALLY REDUCED FORECASTED TAILWINDS; OUR ACFT WAS RUNNING CRITICALLY LOW ON FUEL; THEREFORE THE DECISION WAS MADE TO CLB TO A ALT OF 36500 FT (TO AVOID NAT TRACK ALTS) TO CONSERVE FUEL. ADDITIONALLY; THERE WERE NO VIABLE ALTERNATES FOR LNDG AS CURRENT WX CONDITIONS COULD NOT BE ASCERTAINED DUE TO LACK OF VIABLE COMS ABILITIES AS PREVIOUSLY STATED. WHILE ENRTE AT THIS ALT WE OBSERVED AN AIRLINER THAT APPEARED TO BE INTERSECTING OUR ALT AND HENCE INITIATED A DSCNT AS WE HAD LIMITED ABILITY TO CLB UTILIZING THE TCASII INFO GIVEN. SUBSEQUENTLY; WE CONTINUED AT 28000 FT ON TO SONDRESTROM WHERE THE FLT TERMINATED. SUPPLEMENTAL INFO FROM ACN 631347: WHILE ON TRACK B AT APPROX XA17 GMT AT APPROX N6025.2 W05327.6 AT FL360 I NOTICED TCASII 'POP-UP' TFC ABOUT 20 NM S OF TRACK B HDG IN MY DIRECTION. THE ACFT POPPED UP ON THE SCREEN. WHEN I FIRST NOTICED THE TFC; HE WAS ABOUT 100 FT BELOW MY ALT. THEN TCASII INDICATED THAT THE TFC CLBED TO 400 FT ABOVE MY ALT STILL HEADED DIRECTLY TOWARDS US. AS WE WERE SEARCHING FOR A VISUAL CONTACT THE TCASII TA WENT OFF. AS I DISCONNECTED THE AUTOPLT; WE GOT A TCASII RESOLUTION THAT COMMANDED A DSCNT. WE DSNDED TO COMPLY WITH THE RESOLUTION COMMANDS. AT ABOUT 500-600 FT IN THE DSCNT WE OBTAINED A VISUAL CONTACT AND IT APPEARED THAT THE TFC WAS ALSO IN A DSCNT AND OUR RESOLUTION CHANGED TO A CLBING COMMAND. WE COMPLIED WITH THE INSTRUCTIONS AND THE TARGET PASSED BELOW US AT 500 FT OR MORE ON A PRECISE COLLISION COURSE WITH OUR ACFT. WITH THE EVENT OVER WE RESUMED OUR CRUISING ALT OF 36000 FT AND INFORMED GANDER AND DISPATCH OF THE EVENT. THE TFC APPEARED TO BE A CPR STYLE JET WITH ENGS ON THE TAIL AND APPEARED TO HAVE A BLUE TINT TO THE ACFT. THE SKY WAS CLR WITH A BLUE SKY ABOVE. OUR TRACK WAS ABOUT '280' AND HIS TRACK WAS ABOUT '050. ACFT X WAS BEHIND US AND 1000 FT BELOW US. THEY ALSO RPTED A TCASII TFC ALERT FROM THIS TFC.