
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 the terminal side of angle $\beta$ passes through the point $\left(-2,-1\right)$, angle $\alpha$ is a third quadrant angle, and _____, find the values of the following expressions. Choose one of the following three conditions to fill in the blank line above and complete the solution.<br/>①$tan{2\alpha}=-\frac{4}{3}$; ②$\sin ^{2}\alpha -4\cos ^{2}\alpha =0$; ③$\alpha$ is symmetric to the terminal side of angle $\beta$ with respect to the line $y=x$,<br/>$(1)\frac{{\sin({\frac{\pi}{2}+\alpha})+2\sin({3\pi-\alpha})}}{{4\cos({-\alpha})+\sin({\pi+\alpha})}}$;<br/>$(2)\sin \left(2\alpha +\beta \right)$.

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