As to the quint essential lever gun for deer size game. There can be only one
The winchester model 94 (insert caliber choice)
The model 94 in .30-30 was said to (at one time) taken more whitetail deer
than any other gun and caliber combined.
As for the pump I'd be guessing but the Remington model 7600 (insert caliber)
It seems to have a huge fan base. Toted to have excellent reliability and accuracy, as well as having quick target accusition.
Those are my choices and yes I know, I must be ghey I own 2 94's in .30-30
One was grandpa's and the other was the first rifle I ever bought. I just can't part with either.

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