18. (12 points) Given vectors $\boldsymbol{m}=(\sin A, \cos A)$, $\boldsymbol{n}=(\cos B, \sin B)$, $\boldsymbol{m} \cdot \boldsymbol{n}=\sin 2 C$, and $\angle A$, $\angle B$, $\angle C$ are the angles opposite to sides $a$, $b$, $c$ of $\triangle ABC$ respectively.
(1) Find the size of $\angle C$;
(2) If $\sin A$, $\sin C$, $\sin B$ form a geometric sequence, and $\overrightarrow{C A} \cdot \overrightarrow{C B}=18$, find the value of $c$.
18. (1) From the given information,
$$
\begin{array}{l}
\sin A \cdot \cos B + \cos A \cdot \sin B = \sin 2 C \\
\Rightarrow \sin C = \sin 2 C \Rightarrow \cos C = \frac{1}{2}.
\end{array}
$$

Since $\angle C$ is an interior angle of $\triangle ABC$, we have $\angle C = \frac{\pi}{3}$.
(2) From the given information,
$$
\begin{array}{l}
\sin^2 C = \sin A \cdot \sin B \Rightarrow c^2 = a b. \\
\text{Also, } \overrightarrow{C A} \cdot \overrightarrow{C B} = 18 \text{, then }
\end{array}
$$
$a b \cos C = 18 \Rightarrow c^2 = a b = 36 \Rightarrow c = 6$.