
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 function $f(x)=\cos x(\sin x- \sqrt{3}\cos x)(x∈R)$.

(1) Find the smallest positive period and the decreasing interval of the function $f(x)$.

(2) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $f( \frac{A}{2})=- \frac{ \sqrt{3}}{2}$, $a=3$, $b+c=2 \sqrt{3}$, find the area of $\triangle ABC$.

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