What is the difference between cannot and does not? Would 'does not' be 안 마신다?
Yes, indeed. 못 indicates inability, while 안 is a general negation.
Why is it 마신다 and not something with ~입니다 or ~어요?
What do you mean? 마신다 = written form
Yeah, but how is it formed?
I don't know why they don't add this to the app. I have to use the website to understand stuff.