"to make sure" parallelism and comma

Is the first comma needed in the sentence “To make sure that the audit will go smoothly, and you will avoid future questions from auditors, please …”?

My thinking is that although we normally put a comma if there are two independent clauses linked with “and”, these are not really independent clauses because of the presence of “To make sure”. In other words, “you will avoid future questions” links directly to “To make sure”.

Therefore, the correct punctuation should be “To make sure that the audit will go smoothly and you will avoid future questions from auditors, please …”?

Can you confirm if my thinking is correct?

EDIT:

Well, I don’t know how to show what effort I’ve made to find out the answer myself. Most sources, e.g. this one make it clear that we use a comma when two clauses are joined with “and”, whereas no comma is needed if the two elements are not full clauses. For example in “He slammed the door and never came back.” “never came back” is not a clause as it’s missing the subject. Both “the audit will go smoothly” and “you will avoid problems” are complete clauses as they have their subjects (different in each case), yet both of them are part of a dependent clause starting with “To make sure …”

I was unsuccessful finding any sources that describe the case of independent clauses making up a dependent clause. I don’t even know if my terminology is correct here.

That is, in a nutshell, the reason for me asking this question.