
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 $0 < α < \frac{π}{2}$ and $\frac{π}{2} < β < π$, with $\cos(α + \frac{π}{4}) = \frac{1}{3}$ and $\cos(\frac{π}{4} - \frac{β}{2}) = \frac{\sqrt{3}}{3}$,

1. Find the value of $\cos β$;
2. Find the value of $\cos(2α + β)$.

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