
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 the vectors $m=(\sin x-\sqrt{3}\cos x,1)$ and $n=(\sin (\frac{\pi}{2}+x),\frac{\sqrt{3}}{2})$, and the function $f(x)=m\cdot n$.

(1) Find the interval where $f(x)$ is monotonically increasing.

(2) Given a triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, and given $a=3$, $f(\frac{A}{2}+\frac{\pi}{12})=\frac{1}{2}$, and $\sin C=2\sin B$, find the values of $A$, $c$, and $b$.

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