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I think Bronsted acids and Lewis acids are just different by definitions. Bronsted acids are proton donators and Lewis acids are electron acceptors. I don't think they are related to strong or weak acids.
Whether or not something is an acid or base does not change no matter what definition you use, but the Bronsted/Lewis definitions focus on different characteristics. Bronsted acids donate protons while Bronsted bases accept them. Lewis acids accept electron pairs while Lewis bases donate them. However, an acid is still an acid and a base is still a base no matter what definition you look at.
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