Can I use Project Atlas just before the Runner accesses cards from HQ?

Question

Project Atlas is a Corporation agenda that can be over-advanced to provide agenda counters for the following ability:

Hosted agenda counter: Search R&D for 1 card, reveal it, and add it to HQ. Shuffle R&D.

Because it has no [Click] requirement, I understand that I can use this ability during the Runner's turn.

Imagine that I have a scored Project Atlas with a counter on it and the Runner runs on HQ. They pass the last ICE. Before the Runner accesses HQ, can I use this ability to fetch (say) a Snare into my HQ?

If so, must I do it before they choose to access (so that they could see me add the card and then jack out without accessing)? Or can I do it after they confirm that they want to access a card, giving them no opportunity to jack out?

Answer 1

After reading the FAQ (here), specifically the section on the timing of a run, I believe I can now answer my own question. The relevant steps read:

4 The Runner APPROACHES the attacked server.

4.1 [Paid abilities]

4.2 The Runner decides whether to continue the run.
...Either the Runner JACKS OUT: [end the run unsuccessfully]
...Or the Runner continues the run: go to 4.3.

4.3 [Paid abilities and non-ice can be rezzed]

4.4 The run is considered to be successful. (‘When successful’ conditionals meet their trigger conditions)

4.5 Access cards, then [end the run].

Based on these clarifications, you may use the ability before or after the Runner decides if they'd like to access.

Answer 2

Yes, you can use paid abilities at any point in the timing diagram with the symbol for using paid abilities. There is an option after runner chooses weather to Jack-out or continue to access the server where you can do this. The runner will then have to access your HQ, unless he has used something like Account Siphon.