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

(I) Find the interval(s) where $f(x)$ is monotonically decreasing.

(II) 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 $[0, \frac{π}{2}]$, find $A$, $b$, and the area $S$ of triangle $ABC$.

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