3d Geometry

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 ______________.