
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 $\cos\left(\alpha-\frac{\beta}{2}\right)=-\frac{2\sqrt{7}}{7}$, $\sin\left(\frac{\alpha}{2}-\beta\right)=\frac{1}{2}$, $\alpha\in\left(\frac{\pi}{2},\pi\right)$, and $\beta\in\left(0,\frac{\pi}{2}\right)$. Find the value of $\cos\frac{\alpha+\beta}{2}$ and $\tan(\alpha+\beta)$.

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