
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 the function $f(\alpha)= \dfrac{\cos ( \dfrac{\pi}{2}+\alpha)\cos (2\pi+\alpha)\sin (-\alpha+ \dfrac{3}{2}\pi)}{\sin (\alpha+ \dfrac{7}{2}\pi)\sin (-3\pi-\alpha)}$.

$(1)$ Simplify $f(\alpha)$;

$(2)$ If $\alpha$ is an angle in the third quadrant, and $\cos (\alpha- \dfrac{3}{2} \pi)= \dfrac{1}{5} $, find $f(\alpha)$.

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