I am stuck on a LFT question,

T maps the real axis onto itself and the imaginary axis onto the circle

I do not have a good grasp on LFTs, although I understand fixed points and triples to triples techniques and can solve problems like,

Find a LFT that carries the circle onto the line

But when I have to take into account multiple properties of transformation I get lost, I've read my text a bunch and even picked up another, not to mention the countless number of help articles I've read online. I'm not sure what to do. I've attempted to use fixed points and triples to triples techniques to no useful end. Any hints or reference material would be greatly appreciated!

Thank you!