
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 $$0 < \beta < \frac{\pi}{2} < \alpha < \pi$$, and $$\cos\left(\alpha - \frac{\beta}{2}\right) = \frac{5}{13}$$, $$\sin\left(\frac{\alpha}{2} - \beta\right) = \frac{3}{5}$$.
Find the values of (1) $$\tan\left(\alpha - \frac{\beta}{2}\right)$$;  
(2) $$\cos\left(\frac{\alpha + \beta}{2}\right)$$.

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