Mathematics Question from JEE Main 2023 · 30 January, Shift 1

$\lambda_1 < \lambda_2$ are two values of $\lambda$ such that the angle between the planes $P_1 : \vec{r} \cdot (3\hat{i} - 5\hat{j} + \hat{k}) = 7$ 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 the perpendicular from the point $(38\lambda_1, 10\lambda_2, 2)$ to the plane $P_1$ is _____.