
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 $f(x)=4\sin(\omega x+\varphi)$ ($\omega\gt 0$, $|\varphi|<\frac{\pi}{2}$) passes through the points $(0, 2\sqrt{3})$, $(x_{1}, 0)$, and $(x_{2}, 0)$, and the minimum value of $|x_{1}-x_{2}|$ is $\frac{\pi}{2}$.  
$(1)$ Find the analytical expression of the function $f(x)$ and determine the intervals where the function is monotonically increasing.  
$(2)$ If $f(\alpha)=\frac{12}{5}$, where $\alpha\in\left(\frac{\pi}{12}, \frac{\pi}{2}\right)$, find the value of $\sin 2\alpha$.

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