This puzzle is on the school level
There are two ways to solve this puzzle, but only one will unlock the achievement.
There are also two variations of this puzzle.
Follow the red wires. One of the wires will lead to the library, the other will lead to the looping hallway. Go to the looping hallway.
If you got the first variation, your first cloud of smoke will appear a little over half way down the hallway. If this is the variation you were given, progress forward until you've hit your first cloud of smoke, or until "TURN AROUND" (written on the wall in front of you) disappears.
Now change directions and around the corner you'll see that a new lever has appeared.
Pull the lever to unlock the achievement.
The second variation will have a cloud directly in front of as you open the door. This variation is much trickier because it's pretty buggy. If you were given this variation, I'd recommend bypassing this puzzle and waiting until you're given the first variation on a later play through. You can bypass it by wedging yourself in the doorframe and slowly adjusting the camera until you're able to walk through to the caged room. If you do choose to solve this variation though, loop the hallway until smoke clouds stop appearing, then change direction and pass through the door. Keep going half way through the hall and turning around until the game registers that you've met the criteria and the smoke dissipates. Now pull the new lever and the achievement will pop.