HMMT 2012 T6

It has recently been discovered that the right triangle with vertices $(0,0),(0,2012)$, and $(2012,0)$ is a giant pond that is home to many frogs. Frogs have the special ability that, if they are at a lattice point $(x, y)$, they can hop to any of the three lattice points $(x+1, y+1),(x-2, y+1)$, and $(x+1, y-2)$, assuming the given lattice point lies in or on the boundary of the triangle. Frog Jeff starts at the corner $(0,0)$, while Frog Kenny starts at the corner $(0,2012)$. Show that the set of points that Jeff can reach is equal in size to the set of points that Kenny can reach.