Yes. Of course you can infer. What cannot happen is for your inference to be RAW. What you infer is your ruling or house rule, not RAW.
Ok then it can't be RAW that you "can't do it" either, as you have previously stated, because you need to make an inference to come to the conclusion that it can't be done RAW
The logic you use here is not flawed, but if it is true then it is NOT RAW that you can use action surge in the middle of your attack action and it is also NOT RAW that you can't use action surge in the middle of your attack action.
Essentially there is no RAW concerning this question.
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