OBM 2009 Problem 3

There are $2009$ pebbles in some points $(x,y)$ with both coordinates integer. A operation consists in choosing a point $(a,b)$ with four or more pebbles, removing four pebbles from $(a,b)$ and putting one pebble in each of the points $$(a,b-1),\ (a,b+1),\ (a-1,b),\ (a+1,b)$$

Show that after a finite number of operations each point will necessarily have at most three pebbles. Prove that the final configuration doesn't depend on the order of the operations.