
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 $f(x) = A\sin(\omega x + \frac{\pi}{3})$ ($A>0$, $\omega>0$), the distance between two adjacent axes of symmetry of its graph is $\pi$, and it passes through the point ($\frac{\pi}{3}$, $\frac{\sqrt{3}}{2}$).  
(1) Find the analytical expression of the function $f(x)$.  
(2) If the angle $\alpha$ satisfies $f(\alpha) + \sqrt{3}f(\alpha - \frac{\pi}{2}) = 1$, with $\alpha \in (0, \pi)$, find the value of $\alpha$.

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