
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 $tan\alpha=-\frac{3}{4}$, and $\alpha$ is an angle in the _____ quadrant. Choose one appropriate option from ① first, ② second, ③ third, ④ fourth, and fill in the blank above. Based on your choice, answer the following questions:
- $(1)$ Find the value of $\cos 2\alpha$;
- $(2)$ Simplify and find the value of $\frac{{sin({2\alpha-\frac{\pi}{2}})cos({\frac{3\pi}{2}-2\alpha})tan(2\pi-2\alpha)}}{{tan(-2\alpha-\pi)sin(\pi+2\alpha)}}$.

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