
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 $\omega\ \ \gt 0$, vectors $\overrightarrow{a}=(\sqrt{3}$,$cos^2(ωx+\frac{π}{6})-1)$, $\overrightarrow{b}=(cos(2ωx-\frac{π}{6}),2)$, and the function $f\left(x\right)=\overrightarrow{a}\cdot \overrightarrow{b}$.  
$(1)$ If $\omega =1$, find the interval where $f\left(x\right)$ is monotonically increasing.  
$(2)$ If $f\left(x\right)$ has exactly $3$ zeros in the interval $(0,\frac{π}{2})$, find the range of values for $\omega$.

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