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Put to sleep by anesthesiologist — will get one better? (9)

Ross

I believe the answer is:

outnumber

I'm a little stuck... Click here to teach me more about this clue!

'put to sleep' is the definition.
Although both the answer and definition are verbs in their base form, I don't understand how one could define the other.

'anesthesiologist will get one better?' is the wordplay.
I cannot really understand how this works, but
'get' could be 'be' ('get' can mean 'come to be') and 'be' is located in the answer.
This explanation may well be incorrect...

'by' is the link.

Can you help me to learn more?

(Other definitions for outnumber that I've seen before include "Exceed in total of persons" , "Be more numerous than" , "Exceed numerically" , "Be in the majority" , "Have more (eg troops) than" .)

I've seen this clue in The Hindu.
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