Why does "ablecken" become "abzulecken" in this sentence?

Question

I'm trying to take apart this sentence to understand how it's put together. It's from a children's book called "Wer will eine kleine Katze haben?"

Here is the sentence:

Als Mama die dicke Bella zurücklegt, fängt Mimili an, sie gründlich abzulecken.

I've learned a few things from this sentence that I didn't know, so I'll talk about what I understand and what I don't.

I know that 'als' is a subordinating conjunction, so "Als Mama die dicke Bella zurücklegt" is a subclause, and that's why "zurücklegt" is at the end of it.

Since the subclause came first, the verb of the main clause comes immediately next. However, since the verb is also separable, I put the prefix of the separable verb after the subject rather than at the end of the subclause. This is something I have learned from this sentence.

So far so good. Now the separable verb "anfängt" is ALSO a modal verb, so the verb "ablecken" goes to the end of the main clause.

This all makes sense to me. However, I don't know why "ablecken" transforms to "abzulecken".

Answer 1

When separable prefix verbs are used in an infinitive clause (um...zu, statt...zu etc, ohne...zu etc) the zu is put between the prefix and the main verb

Ich muss weg, um Andreas abzuholen.

Wir gehen angeln, statt an der Besprechung teilzunehmen.

Ich hoffe, bald auszugehen

That's why it's abzulecken, because it's being used in an infinitive clause. In this sentence, however, there isn't anything at the start of the clause to signal it's an infinitive clause (no um, statt, bald, ohne etc), but in English it would also be a similar construction.

Mimili began to lick her thoroughly