
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:
$(1)$ The value of $\frac {\cos (2\pi-\alpha)\cos (\pi+\alpha)\tan ^{2}(2\pi-\alpha)}{\cos ( \frac {\pi}{2}+\alpha)\sin (2\pi-\alpha)\cot ^{2}(\pi-\alpha)}$.
$(2)$ In $\triangle ABC$, $\sin A+ \cos A= \frac { \sqrt {2}}{2}$, $AC=2$, $AB=3$, find the value of $\tan A$.

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