Crossword Genius

Prominent Florentine family shortly finding doctor (5)

Ross

I believe the answer is:

medic

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'doctor' is the definition.
(doctor is a type of medic)

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for medic that I've seen before include "He may offer treatment" , "Eg, doctor" , "Health worker" , "What was Basutoland" , "Physician (informal)" .)

I've seen this clue in The Times.
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