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

1. Find the interval where the function $f(x)$ is monotonically decreasing.
2. Given $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\triangle ABC$, with $A$ being an acute angle, $a=2\sqrt{3}$, $c=4$. If $f(A)$ is the maximum value of $f(x)$ in 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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