Purchasing property with tenant in situ. What do I need to do?

Tenancy Types and Management

Hi,
I am a second-time landlord and have just had an offer accepted on a house with tenant-in-situ. They were on an AST which has timed-out and are now on a rolling tenancy. I have not done this before, so would really appreciate any advice on how to take over the tenancy.

I am happy with the rent they are paying and happy to commit to this for the next 12 months. So would an option be to just begin a brand new AST? Can this just begin on the day of completion, or do they have to have been served notice by their current landlord?

What about just continuing the existing tenancy? Can this be done as I was not a party to that contract?

The tenancy has been managed by a letting agent of good repute, so I am confident they have all the paperwork in order (they are also the selling agent so sure they will pass it on too).

Thanks for any help you can give me!