I'm not a teacher, not even a student.
An action verb is a verb that describes an action, like run, jump, kick, eat, break, cry, smile, or think.
We could think, that "to live" describes an action, and it seems logically possible, but we know,
that "to be" is a linking verb (100%). Actually, "to live" in this sentence is a linking verb too.
We have a rule:
If you can substitute "to be" (am,is, are, was, were, will be etc...) for the verb, that you want to define,
and the sentence still sounds logical, you have a linking verb on your hands.
Many people live in grass-roofed houses.
Many people are in grass-roofed houses. - It makes sense.
It follows - to live is a linking verb.
It's just my opinion, I am not a teacher.