“Having had perfect understanding” or “Having followed all things closely” in Luke 1:3?

Does the phrase mean that Luke had perfect understanding of all things or that he experienced all things in close proximity? The meaning hinges on two words that are translated differently among the translations. First, “παρηκολουθηκοτι” is a participle that indicates a “following” as having occurred. However, “following” in this context seems to refer to a “following of the mind,” which is “understanding.” Thayer's Greek-English Lexicon lists “to understand” as a metaphorical definition of “παρακολουθέω.” Even in English, “follow” is a synonym for “understand” (e.g. “Do you follow my explanation?”). Luke 1:4 indicates that Luke’s intent was that the reader might “know” (επιγνως) with “certainty” (ασφαλειαν). Thus, Luke establishes that he has “understanding” so that he may impart that understanding (knowledge with certainty) to his reader. Secondly, “ακριβως” not only means “closely” but also “accurately.” Thayer defines the word as “exactly, accurately, diligently.” Vine’s Expository Dictionary says it best: “The word expresses ‘accuracy’ which is the outcome of carefulness.” Thus, Luke not only had “close” understanding, but an “accurate” or “perfect” understanding.

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