
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) When $x \in \left[\frac{\pi}{6}, \frac{7\pi}{6}\right]$, find the maximum value of the function $y = 3 - \sin x - 2\cos^2 x$.  
(2) Given that $5\sin\beta = \sin(2\alpha + \beta)$ and $\tan(\alpha + \beta) = \frac{9}{4}$, find $\tan \alpha$.

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