
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 $\alpha \in [\frac{\pi}{4}, \frac{\pi}{3}]$ and $\beta \in [\frac{\pi}{2}, \pi]$, satisfying $\sin(\alpha + \beta) - \sin(\alpha) = 2\sin(\alpha)\cos(\beta)$, find the maximum value of $\frac{\sin(2\alpha)}{\sin(\beta - \alpha)}$.

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