
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 $\cos \alpha = -\frac{1}{3}$, $\sin \beta = \frac{2}{3}$, $\alpha$ is an angle in the third quadrant, and $\beta \in \left(\frac{\pi}{2}, \pi\right)$. Find:
- $(1)$ The value of $\sin 2\alpha$.
- $(2)$ The value of $\cos \left(2\alpha + \beta\right)$.

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