
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 $f\left( \alpha \right)=\frac{\cos \left( \frac{\pi }{2}+\alpha \right)\cdot \cos \left( 2\pi -\alpha \right)\cdot \sin \left( -\alpha +\frac{3}{2}\pi \right)}{\sin \left( -\pi -\alpha \right)\sin \left( \frac{3}{2}\pi +\alpha \right)}$.

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

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