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50how to actually get one
It comes out of the Extractinator, and the per-item odds are grim no matter what you feed in. Desert Fossils are the route worth taking, partly for the slightly better chance and mostly because you can strip an Underground Desert of them far faster than you can shovel up silt or slush. Feed everything, do not sort by hand, and treat it as a background task while you mine.
what the pet does
A baby tyrannosaur walks after you on foot. Outrun it or take to the air and a pterosaur swoops in, grabs it by the tail and carries it along. That handoff is the reason to bother with it.
where the idea comes from
An insect trapped in amber, dinosaur blood inside it, a dinosaur out the other end. It is Jurassic Park, compressed into one drop.