As an ATC I have been taught to acknowledge the TCAS RA Call with 1) Update pertinent traffic information if necessary (types clock reference intentions if known on traffic involved or nearby that the RA may additionally effect and 2)to ask the pilot(s) to report when they receive clear of conflict advice.
I had an RA recently with an aircraft descending to 1000 feet above another (FL330 and FL320), both got the RA and when they had manoeuvred down and up I believe they didn't get clear of conflict until 3400 feet apart and still diverging vertically (FL342 and FL308).
I've also had a 5 way TCAS RA in a holding stack, the top one descending too hard, the next 4 got TCAS RA descend as the one above dived on top of it etc following their RAs.
Both times, I had separation.
TCAS is great security blanket, great for level busts, but very annoying when it happens just because of vertical speed approaching the cleared level.
Personally I think TCAS RA is sufficient, the PAN calls would IMHO be overkill and perhaps just delay getting the right advice.