
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
$(1)$ Given $sin({α+β})=\frac{1}{2}$ and $sin({α-β})=\frac{1}{3}$, find the value of $\frac{{tanα}}{{tanβ}}$;<br/>$(2)$ Given that the obtuse angle $\alpha$ passes through the point $\left(-1,2\right)$, $0 \lt \beta\ \ \lt \pi $, and $cosβ=-\frac{{7\sqrt{2}}}{{10}}$, find the values of $\cos 2\alpha$ and $2\alpha +\beta$.

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