Circuitously is the adverb for circuitous: it tells you how someone goes or how someone speaks — the long way, not the straight way. You travel circuitously if your path loops and doubles back. You answer circuitously if you talk around the point before you say it. Like the adjective, it is a little formal; the long way round or in a roundabout way is what most people say. It does not mean slowly, and it does not mean carefully, though a circuitous explanation can feel slow. The extra time comes from extra turns, not from a gentle speed. You will mostly see this in writing.