
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 $\sin \alpha = -\frac{1}{3}$ and $\alpha$ is an angle in the third quadrant.
(I) Find the value of $\sin 2\alpha$;
(II) Find the value of $\frac{\sin (\alpha - 2\pi) \cdot \cos (2\pi - \alpha)}{\sin^2 (\alpha + \frac{\pi}{2})}$.

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