Is "live" an action verb or a state verb?

I’ve always thought of "live" as a state verb since it implies a state rather than an action. A testament to this is the fact that we cannot use the structure "past time reference + would + live" to talk about a habitual activity in the past (e.g. in "Every summer, we would live in our villa in the country"), instead, we have to use "used to", as in "We used to live in London, but then we moved to Italy."

However, as shown in the picture, the book "Longman Academic Writing" refers to "live" as an action verb (I’m not so sure about "lose" either). I was wondering if the book has made a mistake? Or am I the one in the wrong?