3-day short Romania trip - asking for tips!

Hey there! Hope you're all doing great.

I was hoping to get some tips from locals/experienced people about Romania.
I am in Bucharest for work for 1 day this Saturday but decided to add a few days onto it last minute.

I would have some free time from Sunday morning until Tuesday. Flying out Wednesday.
I don't mind travelling some distance by train. Perhaps rent a car if neccessary.
Are there any national forests/nature relatively close to Bucharest I could go to and spend 2 nights in a simple cabin in nature?
I would love to see some old growth forest and wild life (black bears!?).
I have a lot of experience hiking/travelling so that's not a problem.

Would love to have your advice.
Many thanks and wish you a beautiful day,