
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 $\sin \alpha = -\frac{2\sqrt{5}}{5}$ and $\tan \alpha < 0$,

$(1)$ Find the value of $\tan \alpha$;

$(2)$ Calculate the value of $\frac{2\sin (\alpha+\pi)+\cos (2\pi-\alpha)}{\cos (\alpha - \frac{\pi}{2})-\sin (\frac{3\pi}{2}+\alpha)}$.

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