My suggestion is that "shortly" is an indicator of time, and "briefly" is an indicator of length/duration. This may be a bit confusing, so please look at the examples below.
Ex. I will be at your office shortly (within a short time), so please wait for me.
No: The students replied shortly.
Yes: The students replied briefly. (Their answers were short, perhaps a few words only.)
Ex. I spoke with the doctor briefly (a quick conversation), so I don't have all the details.
No: Are you coming, how long will it take you?
I will be there briefly. (Meaning, I will be there for a small amount of time, but this doesn't answer the question of how long to arrive.)
Yes: Are you coming?
Yes, I will be there shortly, but if you are in a rush, you can go ahead without me.
Hope this helps :)
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