
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 $\tan \alpha$ and $\frac{1}{\tan \alpha}$ are the two real roots of the equation $x^2 - kx + k^2 - 3 = 0$, and $\pi < \alpha < \frac{3}{2}\pi$.

$(1)$ Find the value of $\cos \alpha + \sin \alpha$.

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

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