Three Dimensional Geometry: JEE Main 2023 Mathematics question

If $\lambda_{1} < \lambda_{2}$ are two values of $\lambda$ such that the angle between the planes $P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7$ and $P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9$ is $\sin ^{-1}\left(\frac{2 \sqrt{6}}{5}\right)$, then the square of the length of perpendicular from the point $\left(38 \lambda_{1}, 10 \lambda_{2}, 2\right)$ to the plane $P_{1}$ is ______________.