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 $\mathrm{L}$ 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 $\mathrm{L}$ and $P_{2}$, then $\left(\tan ^{2} \theta\right)\left(\cot ^{2} \alpha\right)$ is equal to ____________.