USAMO 2003 Problem 6

At the vertices of a regular hexagon are written six nonnegative integers whose sum is $2003^{2003}$ . Bert is allowed to make moves of the following form: he may pick a vertex and replace the number written there by the absolute value of the difference between the numbers written at the two neighboring vertices. Prove that Bert can make a sequence of moves, after which the number 0 appears at all six vertices.