
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
The maximum value of the function $f(x) = A\sin(\omega x - \frac{\pi}{6}) + 1$ ($A > 0$, $\omega > 0$) is 3, and the distance between two adjacent axes of symmetry of its graph is $\frac{\pi}{2}$.

(1) Find the analytical expression of the function $f(x)$;  
(2) Find the interval of monotonic increase for the function $y = f(x)$;  
(3) Let $\alpha \in (0, \frac{\pi}{2})$, and $f(\frac{\alpha}{2}) = 2$, find the value of $\alpha$.

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