In a manner that shows excessive self-importance or self-centeredness; in the way an egotist would behave.
From egotistic plus the adverb-forming suffix -ly. This converts the adjective into an adverb describing how actions are performed with egotistical tendencies.
Egotistically describes the action—so while an egotist IS self-centered, they act egotistically, meaning they literally perform self-centeredness in everything they do.
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