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But first they would need to invent some tape
(i.imgur.com)
They could have fitted the whole ring / tape / mouse assembly into a small paper bag Aragorn could have kept it in his jacket and fed it little bits of lembas on the way how lovely x
The ring shrinks down to fit the mouse sized knuckle. After disappearing, the mouse would have hopped on on of those massive eagles and flown away from the volcano.
"so long, suckers!"