Elliptical Cut Design

An elliptical right cylinder has a cross section that is an ellipse with a minor axis extending along the $x$ axis and a major axis extending along the $y$ axis. The semi-minor axis length is $7$ and the semi-major axis length is $15$ . We want to pass an oblique plane through the cylinder such that the resulting intersection between the plane and the cylinder is an ellipse with a semi-minor axis length of $8$ and a semi-major axis length of $16$ . If the unit normal vector to the cutting plane is $\hat{n} = (n_x , n_y, n_z )$ , where $n_x \ge 0 , n_y \ge 0 , n_z \ge 0$ , then find $n_x + n_y + n_z$ .