
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 $5x^2 - 7x - 6 = 0$, find the value of $\frac{\sin \left(\alpha - \frac{\pi}{2}\right)\sin^2\left(-\alpha + \pi\right)\sin \left( \frac{3\pi}{2}-\alpha\right)\tan \left(\alpha - \pi\right)}{\sin \left( \frac{\pi}{2}-\alpha\right)\cos \left(\alpha + \frac{\pi}{2}\right)\cos \left(\pi + \alpha\right)\tan \left(-\alpha + 5\pi\right)}$.

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