
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 $0<\alpha<\pi$ and $0<\beta< \frac {\pi}{2}$, find the value of $\cos(\alpha+\beta)$.

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