Collaborate/Aid/Cooperate/Assist with/in/on something

Hello, everyone!

I would like to know which prepositions to use with each of the following verbs:

Collaborate/Aid/Cooperate/Assist

For example:

I am willing to collaborate/cooperate with the authorities on/in/with the investigation

I am willing to assist/aid the authorities on/in/with the investigation

As far as I can tell, in/on is used when it's followed by a verb, whereas with is used when it's followed by a noun.

P.S: I didn't list help because it's usually followed by with. Is it possible to use it alongside other propositions?