A chemical reaction follows a rate law $v = k[A]^2[B]$, where the rate constant $k$ has units $\text{L}^2 \cdot \text{mol}^{-2} \cdot \text{s}^{-1}$. If the concentration $[A]$ is $0.5 \text{ mol/L}$ and $[B]$ is $0.2 \text{ mol/L}$, and the reaction rate $v$ is $10^{-4} \text{ mol} \cdot \text{L}^{-1} \cdot \text{s}^{-1}$, what is the value of $k$?