The Lost City - Chamber of Exorcism Challenge Tomb (Part 3)

Now for the puzzle. The mechanism in the lower room has 2 rope-wrapped levers which together drain the water from the pool and open the round hatch, causing flammable gas to jet into the room. The hatch only stays open temporarily, however. There's also a cage submerged in the pool, a winch that Lara can crank with her axe, and a rope-wrapped counterweight off to the right.

The first step is to free the cage from the pool. To do this, use rope arrows to pull both levers.

Doing so drains the pool in the floor and opens the round hatch, inundating the room with flammable gas.

Before the hatch closes and water rises again, quickly move down the ramp into the drained pool...

...and use your climbing axe to detach the hook that anchors the cage to the floor.

Return up the ramp and move around to the winch.

Crank it to raise the cage out of the pool.

Now go into the corner to the right of the round hatch.

Turn around and use a rope arrow to pull the counterweight toward you.

This swings the cage around toward the middle of the room. When you have the cage in the correct position, Lara says, "That looks about right."

Move back to the other side of the pool and use rope arrows to pull the 2 levers again. This opens the hatch and fills the room with flammable gas.

As soon as you've pulled the second lever, run up the stairs nearly to the top so you're out of the way of the main gas cloud.

Then shoot a fire arrow into the gas...

...causing an explosion...

...that blasts the cage through the codex room gate.

Now you can enter, pick up various supplies, and examine the codex to unlock GREEK FIRE. This upgrade is equipped automatically. Your normal fire arrows and molotovs now burn blue and are hot enough to melt armor. You also gain XP, and if it's the first time beating the tomb receive 10,000 Expedition Credits.