7. Inside a cylinder with a base radius of 6, there are two spheres with radii also equal to 6, and the distance between their centers is 13. If a plane is made to be tangent to both 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 . $\qquad$
Given, $M C=O_{2} F=6 \Rightarrow \triangle M C D \cong \triangle O_{2} F M$
$$
\Rightarrow M D=O_{2} M=6.5 \Rightarrow M F=2.5 \text {, }
$$

Thus, for the ellipse, $a=6.5, c=2.5, c=6$.
Therefore, the sum of the lengths of the major and minor axes of the ellipse is $2 a+2 b=25$.
Note: The difficulty in this problem lies in the construction of the diagram. Once $\triangle M C D \cong \triangle O_{2} F M$ is identified, the required plane is the one passing through point $M$ and perpendicular to $O_{2} F$.