
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 $\sin \alpha$ is a root of the equation $5x^{2}-7x-6=0$.  
$(1)$ Find the value of $\frac{\sin(-\alpha-\frac{3}{2}\pi) \cdot \cos(\frac{3}{2}\pi-\alpha) \cdot \cos(2\pi-\alpha) \cdot \tan(\pi-\alpha)}{\cos(\frac{\pi}{2}-\alpha) \cdot \cos(\frac{\pi}{2}+\alpha)}$;  
$(2)$ If $\alpha$ is an angle in the fourth quadrant and $\sin(\beta-\frac{\pi}{6})=\frac{5}{13}$ ($0<\beta<\frac{\pi}{2}$), find the value of $\sin(\beta-\alpha+\frac{\pi}{3})$.

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