
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 $\vec{a}=(\sin x, \cos x)$ and $\vec{b}=(2\sqrt{3}\cos x, 2\cos x)$ where $x \in \mathbb{R}$, let $f(x) = \vec{a} \cdot \vec{b}$.

(1) Find the interval(s) where the function $f(x)$ is strictly decreasing.

(2) In triangle $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and the area of the triangle is $S$. If $f(A) = 3$, $B = \frac{\pi}{3}$, and $c = 4$, find the area $S$ of $\Delta ABC$.

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