
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
If $0 < \alpha < \frac{\pi}{2}$, $-\frac{\pi}{2} < \beta < 0$, $\cos \left(\frac{\pi}{4} + \alpha \right) = \frac{1}{3}$, and $\cos \left(\frac{\pi}{4} - \frac{\beta}{2} \right) = \frac{\sqrt{3}}{3}$, then $\cos \left(\alpha - \frac{\beta}{2} \right) = $______.

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