
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
(A-type problem) Given that $α$ and $β$ are within $(0,π)$, and $\tan α=2$, $\cos β =-\frac{7\sqrt{2}}{10}$.

(1) Find the value of $\cos 2α$.

(2) Find the value of $2α-β$.

(B-type problem) Given that $α \in (\frac{\pi}{2}, π)$, and $\sin \frac{α}{2}+\cos \frac{α}{2}=\frac{\sqrt{6}}{2}$.

(1) Find the value of $\cos α$.

(2) If $\sin (α -β)=-\frac{3}{5}$, $β \in (\frac{\pi}{2}, π)$, find the value of $\cos β$.

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