
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 $\overrightarrow{m} = (\sin x, -1)$ and $\overrightarrow{n} = \left( \sqrt{3}\cos x, -\frac{1}{2}\right)$, and the function $f(x) = (\overrightarrow{m} + \overrightarrow{n}) \cdot \overrightarrow{n}$.

$(1)$ Find the interval where $f(x)$ is monotonically decreasing;

$(2)$ Given $a$, $b$, and $c$ are respectively the sides opposite to angles $A$, $B$, and $C$ in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$, and $f(A)$ is exactly the maximum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$, find $A$, $b$, and the area $S$ of $\triangle ABC$.

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