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I must write down his address before I forget.
Literal
Forget-not while [in-に] his address [object-を] note-down-in-advance must.
~ないうちに ('before [it] happens') sets up a window of opportunity — write it down while forgetting hasn't yet set in. ~ておかねばならない stacks three layers: ~ておく (preparation) → おかねば (must, the formal negative conditional of おく) → ならない (obligation). ねばならない is the formal/literary equivalent of なければならない — markedly more bookish, the kind of phrasing you'd expect in essays or formal writing rather than spoken speech.