
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 function $f(x)$ is symmetric about the line $x= \frac {\pi}{6}$, and $\omega\in[0，3]$ when, find the intervals of increase for the function $f(x)$.  
(2) Under the condition of (1), when $x\in[0， \frac {7\pi}{12}]$, the function $f(x)$ has exactly one zero point, find the range of values for the real number $b$.

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