
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 $\overrightarrow{m}=( \sqrt {3}\sin ωx,1)$ and $\overrightarrow{n}=(\cos ωx,\cos ^{2}ωx+1)$, let the function $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}+b$.
(1) If the graph of the function $f(x)$ is symmetrical about the line $x= \frac {π}{6}$ and $ω∈[0,3]$, find the interval where the function $f(x)$ is strictly increasing.
(2) Under the conditions of (1), when $x∈[0, \frac {7π}{12}]$, find the range of real numbers $b$ such that the function $f(x)$ has exactly one zero point.

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