Consider matrices $A \in \mathbb{R}^{n \times m}, B \in \mathbb{R}^{m \times m}$, and $C \in \mathbb{R}^{m \times n}$. Let $r=\operatorname{rank}(A B C)$. Which of the following must be true?
- $\min (m, n) \leq r \leq \max (m, n)$
- $r \leq \min (m, n)$
- $\min (\operatorname{rank}(A), \operatorname{rank}(B), \operatorname{rank}(C)) \leq r \leq \max (m, n)$
- $\min (m, n) \leq r \leq \max (\operatorname{rank}(A), \operatorname{rank}(B), \operatorname{rank}(C))$
- None of the above