
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 $f(x)=\sqrt{3}\sin\omega x\cos\omega x-\cos^2\omega x$ ($\omega>0$), the distance between adjacent symmetric centers of the graph is $\frac{\pi}{2}$.  
$(1)$ Find the intervals where the function $f(x)$ is monotonically increasing.  
$(2)$ If the function $g(x)=f(x)-b$ and $g(x)$ has two zeros in the interval $[0,\frac{\pi}{2}]$, find the range of values for $b$.

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