Example 5 In space, there are four spheres with radii of $2$, $2$, $3$, and $3$. Each sphere is externally tangent to the other three spheres. There is another smaller sphere that is externally tangent to these four spheres. Find the radius of the smaller sphere.
(1995, China Mathematical Olympiad)
Let the radius of the required sphere be $r=\frac{1}{x}$. Then, by Descartes' theorem, we have
$$
\begin{array}{l}
\left(\frac{1}{2}+\frac{1}{2}+\frac{1}{3}+\frac{1}{3}+x\right)^{2} \\
=3\left(\frac{1}{2^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{3^{2}}+x^{2}\right) \\
\Rightarrow\left(\frac{5}{3}+x\right)^{2}=3\left(\frac{13}{18}+x^{2}\right) \\
\Rightarrow x=\frac{11}{6} \Rightarrow r=\frac{6}{11} .
\end{array}
$$