If \( \lambda_{1}<\lambda_{2} \) 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 \( \sin^{-1}\!\left(\dfrac{2\sqrt{6}}{5}\right) \), then the square of the length of perpendicular from the point \( (38\lambda_{1},\,10\lambda_{2},\,2) \) to the plane \( P_{1} \) is \( \_\_\_\_\_ \).
