
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 $f(x)=\sqrt{3}\sin x\cos x+\cos^{2}x$,
(1) Find the interval(s) where the function is strictly increasing;
(2) In $\triangle ABC$, if $f(A)=1$, $\overrightarrow{AB}\cdot\overrightarrow{AC}=4$, and $BC=2\sqrt{3}$, find the lengths of sides $AB$ and $AC$.

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