This sequence is also shown in a gameplay video.
The gold reward is hidden behind those big, brown doors. To open them, begin by grabbing the bottom of the broken stairs on the left and pulling up. Run up the stairs to the broken area to trigger a rolling boulder from above.
Then run off the stairs to the right to avoid being flattened.
To save time, try to jump and grab the dangling rope in the process. If you can't catch the rope, don't worry. Just drop to the floor and wait for a second boulder to pass before climbing back up. Then jump from the staircase to the rope. Adjust Lara's position on the rope and swing over to the staircase on the right.
Run up the stairs just a short way to trigger a third boulder. As soon as you hear the rumbling or see the screen start to shake, either run down the stairs into the small alcove (as indicated by the white arrows in the screenshot above), or keep running up the stairs and duck into the alcove ahead on the right (yellow arrow). You could also jump back to the rope, or just run off the edge near the rope. Once the boulder passes, continue up the flight of stairs on the right.
Stand on the pressure pad to open the door ahead, but don't go through yet. This same pad also opens the big doors below. Stand on the pad until the door in front of you has opened fully. Then you'll know the ones down below are also open. Both doors start to close as soon as Lara steps off the pad.
So run down either set of stairs. . .
. . . hop down to the floor below the rope. . .
. . . and run through the big doors before they close. Once you reach the gold reward (1/1), the lower doors remain open.
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