
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 $f(\alpha)=\frac{{\sin({\frac{\pi}{2}-\alpha})\cos(-\alpha)\tan(\pi+\alpha)}}{{\cos(\pi-\alpha)}}$.  
$(1)$ Simplify $f(\alpha)$;  
$(2)$ If $f(\alpha)=\frac{{2\sqrt{5}}}{5}$, find the value of $\cos \alpha$.

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