
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 $\overrightarrow{a} = (\sin x, -1)$ and $\overrightarrow{b} = (\sqrt{3}\cos x, -\frac{1}{2})$, and the function $f(x) = (\overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{a} - 2$.
(1) Find the smallest positive period and the intervals of monotonic decrease for the function $f(x)$;
(2) Given that $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, 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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