ISL 2007 A7

Let $n>1$ be an integer. In the space, consider the set $$S=\{(x, y, z) \mid x, y, z \in\{0,1, \ldots, n\}, x+y+z>0\}$$ Find the smallest number of planes that jointly contain all $(n+1)^{3}-1$ points of $S$ but none of them passes through the origin. (Netherlands)