AN NMAC OCCURS OFF THE '34 SET' OF RWYS AT SEA; WA.

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CAPT'S CDU MEL'ED INOP; ALL DATA INSERTED AND READ ON FO'S CDU. ACFT AHEAD CLRED TO CROSS RWY 34R; POS AND HOLD RWY 34L. OUR FLT CLRED POS AND HOLD RWY 34R AT INTXN Q SHORTLY THEREAFTER. FMC TKOF DATA BASED ON RWY 34L; SO CAPT ASKED THAT RWY 34R BE INSERTED ON TKOF PAGE. THIS CAUSES 3 THINGS TO OCCUR: 1) REDUCED PWR SETTING REVERTS TO MAX. 2) TKOF SPDS DELETED. 3) INITIAL LNAV LEG LOST LEAVING A LEG DISCONTINUITY. AS WE TOOK THE ACTIVE RWY AT INTXN Q; AND AS I BEGAN LOADING NEW NUMBERS; TWR CLRED US FOR TKOF AS FOLLOWS: 'CLRED FOR IMMEDIATE TKOF TFC ON A 2 1/2 MI FINAL' WHICH I ACKNOWLEDGED; 'CLRED FOR TKOF RWY 34R.' NOW; RATHER THAN LEISURELY TYPING NUMBERS ON A POS AND HOLD CLRNC; WE HAD LIMITED TIME TO LEAVE THE RWY. I CONTINUED TO TYPE DURING THE INITIAL TKOF ROLL AND HAD THE FMC PROGRAMMED BY 1000 FT TKOF ROLL (ESTIMATED). IMMEDIATELY AFTER LIFTOFF; WE GOT AN RA 'MONITOR VERT SPD' AND A STEEP CLB TCASII COMMAND. TFC RESOLUTION INDICATED PRESENT POS 300 FT BELOW US (AT MAX PITCH). AFTER RA WAS RESOLVED; I INFORMED TWR OF THE SIT. THEY INFORMED ME THAT 'ROGER; THAT WAS TFC OFF THE L RWY.' THE CAPT AND I LOOKED AT EACH OTHER SOMEWHAT PERPLEXED; BECAUSE AT NO TIME DID WE EVER HEAR TKOF CLRNC GIVEN TO ACFT AND HAD NO IDEA THERE WAS A POTENTIAL CONFLICT OFF THAT RWY. TWR NEVER INFORMED US OF PRECEDING TFC. OUR CONCERN WAS THAT DUE TO THE FMC CONFUSION; MAYBE WE ASSUMED CLRNC WAS GIVEN TO US WHEN IT WAS ACTUALLY GIVEN TO THE OTHER FLT AND TWR FAILED TO NOTICE THE SIMULTANEOUS TKOF. TO THIS DAY; I HAVE NO IDEA WHAT ACFT CAUSED THE RA BECAUSE WE NEVER WERE ABLE TO VISUALLY ACQUIRE THE TARGET. IT'S VERY PUZZLING BECAUSE THE ONLY TKOF POS TFC CONFLICT I SAW PRIOR TO TKOF WAS SHORT FINAL TO THE L RWY. I NEVER SAW TFC APCHING THE R AND THUS IT APPEARS THE 'CLRED FOR IMMEDIATE TKOF' CLRNC WAS EITHER INTENDED FOR OR ACTUALLY GIVEN TO THE OTHER FLT. FOR ALL I KNOW; HOWEVER; THE OTHER FLT REMAINED ON THE RWY BECAUSE WE NEVER SAW HIM. I READ BACK 'CLRED FOR TKOF;' THE CAPT VERBALLY REPEATED IT; AND WE NEVER HEARD THE OTHER FLT ON THE RADIO. THE MORAL OF THE STORY FOR ME IS TO BE SPRING-LOADED TO STOPPING ALL COMPUTER PROGRAMMING AS THE SIT DEMANDS. SOMETIMES THE SIT CHANGES RAPIDLY AND FLEXIBILITY DEMANDS THE ABILITY TO ADAPT AS REQUIRED. AUTOMATION DEMANDS IMPACTED OUR SITUATIONAL AWARENESS DUE TO THE RAPIDLY CHANGING NATURE OF THE TKOF SCENARIO. WHETHER WE WERE IN ERROR OR NOT; WE DID LOSE TRACK OF THE OTHER ACFT AND THAT IS INEXCUSABLE.