
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 $f(\alpha) = \frac{\sin\left(\alpha - \frac{5\pi}{2}\right)\cos\left(\frac{3\pi}{2} + \alpha\right)\tan\left(\pi - \alpha\right)}{\tan\left(-\alpha - \pi\right)\sin\left(\pi - \alpha\right)}$.
$(1)$ Simplify $f(\alpha)$.
$(2)$ If $\cos\left(\alpha + \frac{3\pi}{2}\right) = \frac{1}{5}$ and $\alpha$ is an angle in the second quadrant, find the value of $f(\alpha)$.

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