
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
Consider the function $f(x)=\cos^2x - \sqrt{3}\sin x\cos x + \frac{1}{2}$.
$(1)$ Find the smallest positive period of $f(x)$.
$(2)$ In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $f(B+C)=\frac{3}{2}$, $a=\sqrt{3}$, and $b+c=3$, find the area of triangle $ABC$.

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