With hearty Jethroe, I have to do well (6)

I believe the answer is:

thrive

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'do well' is the definition.
(to thrive is to succeed or develop well)

I cannot understand how the rest of the clue works.

Can you help me to learn more?

(Other definitions for thrive that I've seen before include "Be successful" , "Grow vigorously, flourish" , "Grow strongly" , "Succeed" , "Hit rev in order to prosper" .)

I've seen this clue in the Evening Standard.

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