Help identifying complete subject

Hello,

This is my first post. My child has a language learning disability, so I work to provide grammar instruction to him, in the hopes that it will help him become more literate. I have no teaching credentials; I merely do my best, as a parent, to instruct.

I am currently using a Holt Elements of Language second course workbook. The workbook does not include the answer key. For one of the exercises I am struggling to identify the complete subject. I am hoping there may be a wise grammarian on this board who can enlighten me.

Here it is: The person looking for information types key words into a search engine.

My son was inclined to identify "The person" as the complete subject because when he saw the verb, "looking", he felt that it must be the start of the predicate. I however, was not convinced. My guess is that the complete subject is: "The person looking for information". I think that my instinct is correct because when I test it by asking who types key words into a search engine, the sensible answer would seem to be: "The person looking for information". I am having trouble understanding how I ought to think about this from a grammatical standpoint, and ultimately, how I should properly instruct my son on analyzing sentences like this.

To me, this phrase (The person looking for information) seems similar to some other examples I might pull out of the air:

-the person in the car
-the person on the bed

These two examples each include a prepositional phrase. The example from the workbook however, would seem to include a verb phrase rather than a prepositional phrase. Is this indeed a verb phrase within the complete subject, and if so, should I now counsel my son that complete subjects may contain verb phrases? This is the first example that I have come across which would seem to include a verb in the complete subject. And unfortunately, I can not find any explanation either in the textbooks I have, nor on the internet, which would help me understand this matter. I must find the answer in order to properly counsel my kiddo.

I would appreciate any knowledgeable feedback. Thanks.