
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 vectors $\overrightarrow{a}=(\sin x,\cos x)$, $\overrightarrow{b}=(\cos x,-\sqrt{3}\cos x)$, and the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$.

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

(2) In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$, respectively. If $f(C)=0$, $0 < C < \dfrac{\pi }{2}$, $c=1$, find the maximum value of the area of $\triangle ABC$.

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