
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 $f(x)=4\sin\frac{ωx}{2}\cos\frac{ωx}{2}-4\sqrt{3}\cos^{2}(\frac{ωx}{2})+2\sqrt{3}$, where $\omega\ \ \gt 0$.  
$(1)$ If the distance between the two adjacent symmetric axes of the graph of the function $f(x)$ is $\frac{π}{2}$, find the analytical expression of the function $f(x)$.  
$(2)$ If the graph of the function $f(x)$ is symmetric about $(\frac{π}{3},0)$ and the function $f(x)$ is monotonic on $[0,\frac{π}{4}]$, find the value of $\omega$.

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