
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 $\sin \alpha = 2 - 4\sin^2\frac{\alpha}{2}$.  
$(1)$ Find the value of $\sin 2\alpha - \cos 2\alpha$.  
$(2)$ Given that $\alpha \in (0, \pi)$ and $\beta \in (\frac{\pi}{2}, \pi)$, and $3\tan^2\beta - 5\tan\beta - 2 = 0$, find the value of $\alpha + \beta$.

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