
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 $\overrightarrow{m}=(\cos\omega x,\sin\omega x)$, $\overrightarrow{n}=(\cos\omega x,\sqrt{3}\cos\omega x)$, and $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}$.  
$(1)$ If the function $y=f(x)$ has a minimum positive period of $2\pi$, find the intervals of monotonicity of $y=f(x)$.  
$(2)$ Given $0 \lt \omega\ \ \lt 2$, if $x=\frac{\pi}{6}$ is a symmetry axis of the graph of the function $y=f(x)$, determine the period and range of $y=f(x)$.

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