HMMT 2010 A7

Let $a, b, c, x, y$, and $z$ be complex numbers such that

$$a=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2} .$$

If $x y+y z+z x=67$ and $x+y+z=2010$, find the value of $x y z$.