
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 $- \frac {\pi}{2} < \beta < 0 < \alpha < \frac {\pi}{2}$, $\cos \left( \frac {\pi}{4}+\alpha\right)= \frac {1}{3}$, $\cos \left( \frac {\pi}{4}- \frac {\beta}{2}\right)= \frac { \sqrt {3}}{3}$, find the value of $\cos \left(\alpha+ \frac {\beta}{2}\right)$.

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