
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 $\alpha \in (0, \frac{\pi}{2})$ and $\beta \in (\frac{\pi}{2}, \pi)$, with $\cos{\beta} = -\frac{1}{3}$ and $\sin{(\alpha + \beta)} = \frac{4 - \sqrt{2}}{6}$.

1. Find the value of $\tan^2{\beta}$.
2. Find the value of $\alpha$.

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