
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
Let the function $f(x) = \sin(\omega x + \varphi)$ ($\omega > 0$, $0 < \varphi < \pi$) have the distance between two adjacent axes of symmetry equal to $\frac{\pi}{2}$. The function $y = f\left(x + \frac{\pi}{2}\right)$ is an even function.  
(1) Find the analytical expression for $f(x)$.  
(2) If $\alpha$ is an acute angle and $f\left(\frac{\alpha}{2} + \frac{\pi}{12}\right) = \frac{3}{5}$, find the value of $\sin 2\alpha$.

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