I think Gandalf has a line, something like, "He's gone to stay with the Elves." Anyway, if we're rewriting, it's easy enough to have Frodo know or be told that Bilbo is in Rivendell. The details of the plan can be figured out; the important changes are that we'd remove the Gandalf travel/research bit, Gandalf would have a more reasonable reason for leaving Frodo and the Ring in the Shire, and Frodo would become a more active character.
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