
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 function $f(x)=\sin(2x+\frac{\pi}{3})-\cos(2x+\frac{\pi}{6})-\sqrt{3}\cos 2x (x\in R)$.

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

(2) In $\triangle ABC$, the acute angle $B$ satisfies $f(B)=\sqrt{3}$, $AC=\sqrt{3}$, and the perimeter of $\triangle ABC$ is $3\sqrt{3}$. Find $AB$ and $BC$.

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