In a way that shows clear and rational thinking; with mental sharpness and good judgment.
From 'clearheaded' plus '-ly' (Old English suffix for forming adverbs). This adverbial form allows the quality of clear thinking to modify actions and decisions.
Using an adverb here transforms a personal trait into a method of *doing* things—it's the difference between 'being' sharp-minded versus 'thinking' your way through a problem step-by-step.
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