
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 $\cos\left(\alpha- \frac{\beta}{2}\right) = -\frac{1}{9}$ and $\sin\left(\frac{\alpha}{2} - \beta\right) = \frac{2}{3}$, with $\frac{\pi}{2} < \alpha < \pi$ and $0 < \beta < \frac{\pi}{2}$, find $\cos(\alpha + \beta)$.

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