The game won't let you even try until you find a piece of paper with some (vague poetic unhelpful) directions. If you don't have that paper, leave the workshop within 5 seconds or you'll be using Second Chance.
To find the paper you need, go to the Tartarus area of the Underworld (the one with the pool). Stand with your back to the red "hands" door and walk forward, then right. The paper you need is on the steps going up to the right.
Now you can go back to the Workshop and tackle that alarm. Read to the end of this hint for a picture showing how to shut off the alarm.
Looking at the alarm panel, you see circles covered with smudgy fingerprints that almost show a path, but not quite. Hmm.
The paper, which is labeled "Workshop Notes" in your inventory, says ... well, the important thing it says is that "the artist leaves fingerprints" and "all circles connect."
That suggests that the panel has been traced with a finger, and also that whatever you do here, you need to end up where you started.
But you can't trace any old path; you have to trace the specific path shown by the fingerprints. And you have to touch every circle before you return to your starting point.
Start anywhere and see how far you get. Be careful not to block off your starting point; remember, you have to end up there eventually.
Once you start tracing, you have a couple minutes to finish the path, but don't dawdle. I had to lose and redo it several times before I got it.