Bet Italian smoker lapses back (4)

I believe the answer is:

ante

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This could be a double definition.

'bet italian smoker lapses' is the first definition.
'ante' can be an answer for 'bet' (I've seen this before). I am unsure of the 'italian smoker lapses' bit.

'back' is the second definition.
(thesaurus)

Can you help me to learn more?

(Other definitions for ante that I've seen before include "Stake, bet" , "not post" , "Poker pot starter" , "Stake (that may be upped?)" , "Pay" .)

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