
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 vectors $\overrightarrow{m}=(\sqrt{3}\sin \omega x,1)$ and $\overrightarrow{n}=(\cos \omega x,\cos^2 \omega x+1)$, let the function $f(x)=\overrightarrow{m}\cdot \overrightarrow{n}+b$.

$(1)$ If the graph of the function $f(x)$ is symmetric about the line $x=\frac{\pi}{6}$, and $\omega \in [0,3]$ when, find the intervals of monotonic increase for the function $f(x)$;

$(2)$ Under the condition of $(1)$, when $x\in \left[0,\frac{7\pi}{12}\right]$, the function $f(x)$ has exactly one zero point, find the range of the real number $b$.

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