Je ne pense pas que je puisse le faire

That's an interesting question, Barbara. Cécile and I were discussing that rule in the replies to this question: https://progress.lawlessfrench.com/questions/view/2-subjects-subjonctif

I think there is such a rule, but penser and croire may be exceptions. Perhaps that's because they are often followed by the indicative, and it would be odd if you could say "je pense que je peux le faire" but not "je ne pense pas que je puisse le faire".

Among people who know more about these things than I do, there are differences of opinion. If you look at the following link and scroll down to the very bottom, Laura answers a question on exactly this example:

https://www.lawlessfrench.com/grammar/subjunctive/

Laura argues that you can't use the subjunctive and so you have to say "Je ne pense pas pouvoir le faire"

However in a thread on WordReference, the user "Maitre Capello" discusses a similar example using croire. https://forum.wordreference.com/threads/fr-ne-pas-croire-que-mode.643348/ He says:

Actually, Je ne crois pas que je puisse and Je ne crois pas pouvoir are both fine, but I prefer the former. (Note: I'm only talking about ne pas croire specifically. With some other verbs you have no choice but use an infinitive clause.) 

There's also a very useful thread here:

https://french.stackexchange.com/questions/39733/subjunctive-mood-when-the-the-subject-is-the-same-in-main-and-dependent-clauses