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How do you get nice corners where multiple rounded edges…

How do you get nice corners where multiple rounded edges meet?

Absolute beginner, so I apologize in advance for not knowing any of the right words :)

I'm trying to make a shallow, open ended tray (picture a very wide U-shaped cross section, with no front or back). The plan is to put some legs on it and an sit my old electric keyboard in there, so it has a slightly nicer looking wooden stand. I'm thinking this is effectively just a keyboard-sized board, with a short left wall and right wall butt-jointed on.

Making this tray seems simple enough, but I then want to round the edges off so there are no harsh 90 degree angles. I have distant memories of something called a roundover bit, and I see how I could use that to clean up a single 90 degree edge. What I'm worried about is where multiple of these rounded edges meet. I feel like this will necessitate some kind of vaguely spherical corner profile and I don't see how a router will be able to achieve that when it only follows one linear axis at a time.

You can see an example on this chair here https://i.imgur.com/FRFG3xz.png , where the cushion of the chair is sitting inside a big U-shaped tray. The front edge is rounded, the face of each arm is rounded, the corners on the sides are rounded... but all of these planes must meet and somehow resolve at a single point. Is this just a file and a lot of patience?

I would also be open to glueing bits of rounded trim (bullnose beads?) on top of the basic structure but it has the same problem - I expect I would need some kind of corner piece, and I don't know what that's called.

I've made a lot of shelves and cupboards and storage boxes over the years, so I'm very good at cutting things into rectangles and joining them together, I've just never tried to make something look nice. I am sure this is going to be a very humbling project :)
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