
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 $f(x)=(\overrightarrow{a}+\overrightarrow{b})\cdot\overrightarrow{a}-2$.
1. Find the interval of monotonic increase for the function $f(x)$;
2. Given $a$, $b$, $c$ are respectively the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, 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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