AN ACR CREW BECOMES INVOLVED IN THE TRACK FOLLOWED BY AUTOFLT AND NEGLECTED THE MONITORING OF THE REQUIRED ALT DURING THE BEGINNING OF THE APCH TO AN ARPT IN MOUNTAINOUS TERRAIN.

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WE HAD ENTERED HOLDING AT RLG AWAITING OUR TURN TO START THE VNAV LOC RWY 25 APCH INTO EGE. WE HAD JUST PASSED RLG WHEN CLRED FOR THE APCH. CAPT HAD ME PUNCH IN 'EXIT HOLD' FUNCTION ON FMC. FJC SWITCHED TO A CIRCLE (VICE HOLDING PATTERN) TO REACH RLG. CAPT AND I BEGAN FOLLOWING APCH PROCS AS LISTED ON THE DESCRIPTION PAGE OF THE APCH. ACFT AS DISPLAYED ON MAP MODE CIRCLED THE RLG FIX; BUT NEVER SEEMED TO GET TO RLG. ENGAGED LNAV AND VNAV ACCORDINGLY; SET 7800 FT IN THE ALT WINDOW AS DICTATED BY APCH. MEANWHILE ACFT HAD PASSED N OF RLG AND PASSED E OF THE VOR WHILE TURNING S; ACFT BEGAN TO DSND TOWARD 13100 FT. (ACFT ACTUALLY NEVER FLEW DIRECTLY OVER RLG). FMC CONTINUED TO DISPLAY A CIRCLE INTO THE FIX RLG AND AS ACFT REACHED SE QUADRANT; THE CAPT HAD ME SELECT 'HEADING SELECT' FUNCTION ON THE FLT CTL MODE PANEL. AS WE PASSED W OF THE 184 DEG RADIAL (FROM RLG); CAPT HAD ME PUT A 'DIRECT TO' FUNCTION IN THE FMC AND WENT BACK TO LNAV. WE DECIDED THE ACFT WOULD NOT BE ABLE TO MAKE THE 180 DEG TURN (AS WE WERE NOW TURNING N) BACK TO THE 184 DEG RADIAL; SO CAPT WENT BACK TO HEADING SELECT TO WIDEN THE TURN IN ORDER TO MAKE THE INTERCEPT EASIER. ACFT CONTINUED TO DSND THROUGHOUT TURN AND ATC THEN QUERIED US ABOUT OUR LOCATION AND ALT. ON SECOND TIME OVER RLG LNAV CAPTURED THE 184 DEG RADIAL AND ACFT FINALLY FOLLOWED THE APCH. IN HINDSIGHT; WE SHOULD HAVE REQUESTED ANOTHER TURN IN HOLDING TO ALLOW FOR A BETTER ENTRY INTO THE APCH. I FAILED IN MY JOB AS I WAS WRAPPED UP IN WATCHING THE 'VIDEO GAME' AND FORGOT TO CONSIDER AIRSPACE AND TERRAIN CLRNC. I WOULD ESTIMATE WE STAYED WITHIN 3-4 MI AT ALL TIMES AROUND THE RLG VOR.