1. Inside a cylinder with a base radius of 6, there are two spheres with radii also of 6, and the distance between their centers is 13. If a plane is made to be tangent to these two spheres and intersects the cylindrical surface to form an ellipse, then the sum of the lengths of the major and minor axes of this ellipse is $\qquad$
1. Through the point of tangency between the plane and the sphere, construct the axial section of the cylinder (as shown in the figure). Clearly, the minor axis of the ellipse is the diameter of the cylinder, i.e., $2 b=12$.
$$
\begin{array}{ll}
\text { Also, } & \cos \alpha=6: \frac{13}{2}=\frac{12}{13}, \\
\therefore & a=\frac{6}{\cos \alpha}=\frac{13}{2}, \\
\therefore & 2 a=13,2 a+2 b=25 .
\end{array}
$$