
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 $f(\alpha) = \frac{\sin(\alpha - \frac{\pi}{2})\cos(\frac{3\pi}{2} + \alpha)\tan(\pi - \alpha)}{\tan(-\alpha - \pi)\sin(-\alpha - \pi)}$, where $\alpha \neq \frac{1}{2}k\pi$ ($k \in \mathbb{Z}$).
(1) Simplify $f(\alpha)$;
(2) If $f(\frac{\pi}{2} + \beta) = -\frac{\sqrt{3}}{3}$, where $\beta$ is an angle in the fourth quadrant, find the value of $\sin(2\beta + \frac{\pi}{6})$.

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