
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 $0 < β < \dfrac{π}{2} < α < π$, and $\cos (α- \dfrac{β}{2} )= \dfrac{5}{13} $, $\sin ( \dfrac{α}{2}-β)= \dfrac{3}{5} $. Find the values of:
$(1) \tan (α- \dfrac{β}{2} )$
$(2) \cos ( \dfrac{α+β}{2} )$

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