
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 $f(x)=\sqrt{3}\sin \omega x\cos \omega x-\cos ^{2}\omega x+m$.  
$(1)$ If the graph of $f(x)$ is symmetric about the line $x=\frac{2\pi}{9}$, where $\omega \in [1,\pi]$, find the interval where $f(x)$ is monotonically increasing.  
$(2)$ Under the condition in $(1)$, when $x\in [0,\frac{\pi}{2}]$, let $x_{1}$ and $x_{2}$ be two zeros of $f(x)$. Find the value of $f(x_{1}+x_{2})-m$ and the range of $m$.

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