
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 < a < \frac{π}{2}, - \frac{π}{2} < β < 0, \cos \left(a+ \frac{π}{4}\right)= \frac{1}{3} \sin \left( \frac{π}{4}-\frac{β}{2}\right)= \frac{ \sqrt{3}}{3} $, find $\cos \left(a+ \frac{β}{2}\right)$.

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