KöMaL 2008 A448

The numbers 1,2,...,$N$ are colored with 3 colors such that each color is used at most $\frac{N}{2}$ times. Let $A$ be set of all (ordered) 4-tuples ($x$,$y$,$z$,$w$), consisting of such numbers, such that $x+y+z+w\equiv 0 \pmod{N}$ and $x$, $y$, $z$, $w$ have the same color. Similarly, let $B$ be the set of all (ordered) 4-tuples ($x$,$y$,$z$,$w$) such that $x+y+z+w\equiv 0 \pmod{N}$ , the numbers $x$, $y$ have the same color, $z$ and $w$ have the same color, but these two colors are distinct. Prove that |$A$|$\le$|$B$|.