
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 $cos({α-\frac{β}{2}})=-\frac{{2\sqrt{7}}}{7},sin({\frac{α}{2}-β})=\frac{1}{2},\frac{π}{2}＜α＜π,0＜β＜\frac{π}{2}$. Find the value of $(1)$ $cos\frac{{α+β}}{2}$; $(2)$ $\tan \left(\alpha +\beta \right)$.

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