
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 the function $f(x) = A\sin(2x + \phi) (A > 0, 0 < \phi < \frac{\pi}{2})$, when $x = \frac{\pi}{12}$, the maximum value of $f(x)$ is $2$.
(1) Find the smallest positive period and the expression of $f(x)$;
(2) If $f(\alpha + \frac{\pi}{3}) = -\frac{1}{2}, \alpha \in [0, \frac{\pi}{4}]$, find the value of $f(\alpha + \frac{\pi}{6})$.

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