
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 and $f(\alpha)= \dfrac{\sin (\pi-\alpha)\cos (2\pi-\alpha)\tan (-\alpha+3\pi)}{\cos (-\alpha-\pi)\sin (-\alpha-\pi)}$

$(1)$ Simplify $f(\alpha);$  
$(2)$ If $\cos \left(\alpha -\dfrac{3\pi }{2}\right)=\dfrac{1}{5}$, find the value of $f(\alpha);$  
$(3)$ If $\alpha =-{{1860}^{{}^{\circ }}},$ find the value of $f(\alpha).$

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