
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 $\tan\alpha = -\frac{3}{2}$, where $\alpha$ is an angle in the second quadrant  
$(1)$ Find the value of $\frac{\sin(-\alpha - \frac{\pi}{2})\cos(\frac{3}{2}\pi + \alpha)\tan(\pi - \alpha)}{\tan(-\alpha - \pi)\sin(-\pi - \alpha)}$;  
$(2)$ Find the value of $\frac{1}{\cos\alpha\sqrt{1 + \tan^2\alpha}} + \sqrt{\frac{1 + \sin\alpha}{1 - \sin\alpha}} - \sqrt{\frac{1 - \sin\alpha}{1 + \sin\alpha}}$.

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