Three-gap problem, easy version.

Question

Let $N$ be a positive integer and $\theta$ an angle in $(0, 2\pi)$. Consider the map$$f: \{0, 1, 2, \dots, N-1, N\} \to \text{unit circle}, \text{ }f(k) = k\theta \text{ }(\text{mod } 2\pi).$$Show that the image of $f$ divides the circle into arcs of $1$, $2$, or $3$ different lengths.