$5 、$ Given that two large spherical surfaces with a radius of 1 are externally tangent to each other, and both are internally tangent to a cylindrical surface with a radius of 1, another smaller spherical surface is externally tangent to both large spherical surfaces and internally tangent to the cylindrical surface. A plane passing through the centers of the small and large spheres intersects the cylindrical surface to form an ellipse. Then the maximum possible value of the eccentricity of this ellipse is $\qquad$
5、 $\frac{4}{5}$.
It is easy to calculate that the radius of the small sphere is $\frac{1}{4}$. Let the line passing through the center of the small sphere and the center of a large sphere intersect the cylinder at points $C$ and $D$. Then, the section of the cylinder by the plane through $C$ and $D$ is an ellipse with a constant major axis length $CD$, and it is easy to know that $CD=\frac{10}{3}$. The minimum value of the minor axis length is the diameter of the base, which is 2. Therefore, the eccentricity of the ellipse $e=\sqrt{1-\left(\frac{b}{a}\right)^{2}} \leqslant \sqrt{1-\left(\frac{3}{5}\right)^{2}}=\frac{4}{5}$.