
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 $f(x)=\frac{\sqrt{3}}{2}\sin (2x+\frac{\pi }{3})-\cos^{2}x+\frac{1}{2}$.

(1) Find the interval(s) where the function $f(x)$ is monotonically increasing on $[0,\pi]$;

(2) In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$, respectively. If $f(A)=\frac{1}{4}$ and $a=3$, find the maximum area of $\triangle ABC$.

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