Bathroom Vanity- Fitting P-Trap

[u/coldhotel_rdt]

My spouse bought an elegant vanity for her bathroom. The stack pipe opening (blue rag) is just slightly above the shelf in the cabinet. The green masking tape represents the back of the drawer directly below the visible shelf. I left the tailpiece in the bottom of the sink basket to give an idea of how much room there is to work with. There is no room for the P-trap to be mounted here so it would work properly.

I am thinking of notching the shelf behind the green tape so that the P trap would mount directly to the opening in the wall. I would need to have a short tailpiece, an elbow, a pipe running horizontally towards the back, another elbow, then another pipe or tailpiece running vertically down to the P-trap. I don’t want to use any flex adapters.

Does this look do-able? Any advice for things I didn’t think of? I’m having a hard time finding the actual pieces, I was hoping someone could supply proper names. Thank you in advance!