
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 a function $f(\alpha)=\frac{\cos(\frac{\pi}{2}+\alpha)\cos(2\pi+\alpha)\sin(-\alpha+\frac{3}{2}\pi)}{\sin(\alpha+\frac{7}{2}\pi)\sin(-3\pi-\alpha)$.<br/>$(1)$ Simplify $f(\alpha)$ and find the value of $f(1230^{\circ})$;<br/>$(2)$ If $\alpha$ is an angle in the third quadrant and $\cos(\alpha-\frac{3}{2}\pi)=\frac{1}{5}$, find $f(\alpha)$.

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