
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 $2\sin \alpha =2\sin ^{2}\frac{\alpha}{2}-1$.  
$(1)$ Find the value of $\sin 2\alpha +\cos 2\alpha$;  
$(2)$ Given $\alpha \in (0,\pi)$, $\beta \in (\frac{\pi}{2},\pi)$, $3\tan ^{2}\beta -2\tan \beta =1$, find the value of $\alpha +\beta$.

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