
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 that $x\in R$, vector $\overrightarrow{OA}=\left(2a{\cos }^{2 }\frac{\omega x+\phi }{2},1\right)$, $\overrightarrow{OB}=\left(1,\sqrt{3}a\sin \left(\omega x+\phi \right)-a\right)$, and the function $f(x)=\overrightarrow{OA}\cdot \overrightarrow{OB}$ where $(a\neq 0,\omega >0,0<\phi <\frac{\pi }{2})$, the distance between the adjacent two highest points of the graph of $f(x)$ is $\pi$, and its graph has a symmetrical axis equation of $x=\frac{\pi }{12}$.

(1) Find the expression for the function $f(x)$;
(2) Find the interval(s) where $f(x)$ is monotonically increasing when $a > 0$;
(3) Find the values of $a$ and $b$ when the maximum value of $f(x)+b$ is $2$ and the minimum value is $-\sqrt{3}$ on the interval $x\in [0,\frac{\pi }{2}]$.

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