
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 $a=(\sin x,-1)$ and $b=( \sqrt{3}\cos x,- \frac{1}{2})$, and the function $f(x)=(a+b)·a-2$.

(1) Find the smallest positive period $T$ of the function $f(x)$.

(2) Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$ respectively, where $A$ is an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)=1$, find the area $S$ of $\triangle ABC$.

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