
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 $f(x)=\sqrt{3}\sin\omega x-\cos\omega x$, where $\omega > 0$.
$(1)$ If the distance between two adjacent symmetric axes of the graph of function $f(x)$ is $\pi$, find the interval where $f(x)$ is monotonically increasing.
$(2)$ If the graph of function $f(x)$ is symmetric about $({\frac{\pi}{2},0})$ and $f(x)$ is monotonically increasing on $[{0,\frac{\pi}{3}}]$, find the value of $\omega$.

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