
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
$(1)$ Given the equation $\sin \left(\alpha -3\pi \right)=2\cos \left(\alpha -4\pi \right)$, find the value of $\frac{{\sin({\pi-\alpha})+5\cos({2\pi-\alpha})}}{{2\sin({\frac{{3\pi}}{2}-\alpha})-\sin({-\alpha})}}$. <br/>$(2)$ Given $-\frac{\pi}{2} < x < 0, \sin x + \cos x = \frac{1}{5}$, find the value of $\sin x - \cos x$.

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