
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 $\overrightarrow{a}=(\sin x,-1)$, $\overrightarrow{b}=( \sqrt{3}\cos x,- \frac{1}{2})$ and the function $f(x)=( \overrightarrow{a}+ \overrightarrow{b})· \overrightarrow{a}-2$.
(1) Find the interval where the function $f(x)$ is monotonically increasing;
(2) Given $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ of $\triangle ABC$ respectively, where $A$ is an acute angle, $a= \sqrt{3}$, $c=1$, and $f(A)=1$, find the area $S$ of $\triangle ABC$.

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