Let $x,$ $y,$ and $z$ be nonzero real numbers, such that no two are equal, and
\[x + \frac{1}{y} = y + \frac{1}{z} = z + \frac{1}{x}.\]
Find all possible numeric values of $xyz.$
Try letting x = a and y = 1/b. Then you should get the result that xyz can be -2, -1, 1 or 2.
