
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given vectors $\overrightarrow{a} = (\cos \alpha, \sin \alpha) (0 \leqslant \alpha < 2\pi)$ and $\overrightarrow{b} = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$.
(1) If $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$, find the value of $\alpha$;
(2) If the vectors $\sqrt{3} \overrightarrow{a} + \overrightarrow{b}$ and $\overrightarrow{a} - \sqrt{3} \overrightarrow{b}$ are perpendicular, find $\tan \alpha$.

Present the answer in LaTex format: \boxed{Your answer}
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