Pre-Algebra · Superchallenge problem

A path visits the lattice points with nonnegative coordinates in this order:

$(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$, $(0,2)$, $(1,2)$, $(2,2)$, $(2,1)$, $(2,0)$, $(3,0)$, $(3,1)$, $(3,2)$, $(3,3)$, $(2,3)$, $(1,3)$, $(0,3)$, $(0,4)$, $\ldots$

Each time it has visited every point of a square block from $(0,0)$, it steps out and covers the next L-shaped layer, snaking back and forth in the pattern shown. What is the sum of the coordinates of the $2000$th point on the path?