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

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