Let $\theta$ be the angle between the planes $P_1: \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2: \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$. Let $L$ be the line that meets $P_2$ at the point $(4,-2,5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between $L$ and $P_2$, then $\tan^2\theta \cdot \cot^2\alpha$ is equal to $\underline{\hspace{1cm}}$.