$\triangle ABC$ is a right triangle with $\angle C$ as the right angle. One ant starts at point $A$, and another ant starts at point $B$. The ants travel the vertices in the order $A \rightarrow B \rightarrow C \rightarrow A$ in an endless cycle. The midpoint of the line segment connecting the two ants forms a closed region $R$. Find the maximum possible value of $\frac{[R]}{[\triangle ABC]}$.