
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 $0 < α < \frac{\pi }{2} < β < π$, $\tan \frac{α}{2}= \frac{1}{2}$, and $\cos (β-α)= \frac{ \sqrt{2}}{10}$, find:

(1) The value of $\frac{\cos 2\alpha }{\sqrt{2}\cos (\dfrac{\pi }{4}+\alpha )\sin \alpha }$

(2) The value of $β$

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