USAMO 2007 — Days 1 and 2 Question 4

An animal with $n$ cells is a connected figure consisting of $n$ equal-sized square cells (equivalently, a polyomino with $n$ cells). A dinosaur is an animal with at least $2007$ cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.

Proposed by Reid Barton; source discussion: https://aops.com/community/p825505