
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 $0 < \alpha < \frac{π}{2}$, $-\frac{π}{2} < \beta < 0$, $cos({\frac{π}{4}+α})=\frac{1}{3}$, $cos({\frac{π}{4}-\frac{β}{2}})=\frac{{\sqrt{3}}}{3}$. Find the following:
1. The value of $cos({α+\frac{β}{2}})$.
2. The value of $\sin \beta$.
3. The value of $\alpha -\beta$.

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