13. (25 points) Given that $a$, $b$, and $c$ are the sides opposite to the internal angles $\angle A$, $\angle B$, and $\angle C$ of $\triangle ABC$, respectively, and
\[ b \cos C + \sqrt{3} b \sin C - a - c = 0 \].
(1) Prove that $\angle A$, $\angle B$, and $\angle C$ form an arithmetic sequence;
(2) If $b = \sqrt{3}$, find the maximum value of $2a + c$.
13. (1) From the given condition, we have
$$
\sin B \cdot \cos C+\sqrt{3} \sin B \cdot \sin C-\sin A-\sin C=0 \text {. (1) }
$$

Since $\angle A+\angle B+\angle C=\pi$, then $\sin A=\sin (B+C)$.
Substituting the above into equation (1) yields
$$
\begin{array}{l}
\sin B \cdot \cos C+\sqrt{3} \sin B \cdot \sin C- \\
(\sin B \cdot \cos C+\cos B \cdot \sin C)-\sin C=0 \\
\Rightarrow \sqrt{3} \sin B \cdot \sin C-\cos B \cdot \sin C-\sin C=0 .
\end{array}
$$

Since $0 < \angle B < \pi$, we have $\sin C > 0$.
Thus, $\sqrt{3} \sin B-\cos B-1=0$
$$
\begin{array}{l}
\Rightarrow 2\left(\sin B \cdot \cos \frac{\pi}{6}-\cos B \cdot \sin \frac{\pi}{6}\right)=1 \\
\Rightarrow 2 \sin \left(B-\frac{\pi}{6}\right)=1
\end{array}
$$

Since $0 < \angle B < \pi$, we have $\angle B=\frac{\pi}{3}$.
From $\angle A+\angle B+\angle C=\pi$, we know
$$
\angle A+\angle C=\frac{2 \pi}{3}=2 \angle B \text {. }
$$

Therefore, $\angle A, \angle B, \angle C$ form an arithmetic sequence.
(2) By the Law of Sines, we have
$$
\begin{array}{l}
\frac{a}{\sin A}=\frac{c}{\sin C}=\frac{\sqrt{3}}{\frac{\sqrt{3}}{2}} \\
\Rightarrow a=2 \sin A, c=2 \sin C \\
\Rightarrow 2 a+c=4 \sin A+2 \sin C .
\end{array}
$$

Since $\angle B=\frac{\pi}{3}$, we know $\angle C=\frac{2 \pi}{3}-\angle A$.
Thus, $2 a+c=4 \sin A+2 \sin \left(\frac{2 \pi}{3}-A\right)$.
$$
\begin{array}{l}
=4 \sin A+2\left(\frac{\sqrt{3}}{2} \cos A+\frac{1}{2} \sin A\right) \\
=5 \sin A+\sqrt{3} \cos A \\
=2 \sqrt{7} \sin (A+\varphi) \leqslant 2 \sqrt{7},
\end{array}
$$

where the acute angle $\varphi$ satisfies
$$
\sin \varphi=\frac{\sqrt{21}}{14}, \cos \varphi=\frac{5 \sqrt{7}}{14} .
$$

Since $\angle A \in\left(0, \frac{2 \pi}{3}\right)$, when $\angle A$ is the complement of $\varphi$, $2 a+c$ reaches its maximum value of $2 \sqrt{7}$.