(7) An ant starts from vertex $A$ of the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, and travels along the surface of the prism to reach vertex $C_{1}$. The shortest distance is 6. Then, the maximum volume of the rectangular prism is $\qquad$ .
(7) $12 \sqrt{3}$ Hint: Let the lengths of the three edges starting from the same vertex of the cuboid be $a$, $b$, and $c$, and $a \leqslant b \leqslant c$. Then, the shortest distance from $A$ to vertex $C_{1}$ along the surface of the cuboid is $\sqrt{(a+b)^{2}+c^{2}}=6$, so $(a+b)^{2}+c^{2}=36$.
Since
$$
\begin{aligned}
(a+b)^{2}+c^{2} & =a^{2}+b^{2}+2 a b+c^{2} \\
& \geqslant 2 a b+2 a b+c^{2} \\
& \geqslant 3 \sqrt[3]{4 a^{2} b^{2} c^{2}}
\end{aligned}
$$

Therefore, $4 a^{2} b^{2} c^{2} \leqslant 12^{3}$, so the volume of the cuboid $V=a b c \leqslant 12 \sqrt{3}$. The equality holds when $a=b$, $2 a b=c^{2}$, i.e., $a=\sqrt{6}$, $b=\sqrt{6}$, $c=2 \sqrt{3}$.