
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 that $\pi < \alpha < 2\pi$ and $\cos(\alpha - 7\pi) = -\frac{\sqrt{3}}{2}$,

(1) Find the value of $\sin(3\pi + \alpha) \cdot \tan(\alpha - \frac{7\pi}{2})$;

(2) If vectors $\vec{a} = (\sin\alpha, \cos x)$ and $\vec{b} = (-\sin x, \cos\alpha)$, and $\vec{a} \perp \vec{b}$ with $x \in (0, \pi)$, find the real number $x$.

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