Ready pistols and take a running jump from the walkway near the entrance onto the slightly higher walkway to the east.
Kill the rat (8) lurking there, then walk down the steps and to the right.
Grab the horizontal crevice above and traverse to the left until you can drop down onto the corner ledge with the shotgun shells (2).
Grab the crevice on the other side of the corner, continue traversing all the way to the left, then drop onto the elevated walkway. (If you like, you can take a detour along the top of the L-shaped corner ledge to get the RUSTY KEY, but to save backtracking, we'll wait and get it later.)
Turn around, go to the other end of this walkway, and climb onto the square pillar at the corner. (There's a save crystal here in New Game+.) From the top of the pillar, take a standing jump onto the high balcony.
Your buddy Pierre is waiting inside. To get rid of him quickly and easily, draw the shotgun and run up the stairs. As you enter the room, turn right...
...and blast Pierre twice at close range when he emerges from behind the square pillar.
Then run/jump past him.
He should scurry out the door and disappear. You can use pistols here if necessary, but Lara will probably lose some health in the process. In any case, try to injure him and then get behind him as quickly as possible. If you block his route to the exit, he'll keep shooting at Lara.
Don't use the switch yet. It floods the cistern, and you still need to do a few things above water. If you've already done this, move it back to the up position to drain the water.
DARK MODE: Even with brightness adjusted, dark images on a light background can be hard to see. May I suggest a reputable dark mode browser extension, such as Dark Reader? (Download only from trusted sites, like darkreader.org or your browser's web store.)
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