
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
Let the function $f(x) = \cos^2x - \sqrt{3}\sin x\cos x + \frac{1}{2}$.

$(1)$ Find the smallest positive period and the range of $f(x)$;  
$(2)$ Given in $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(B+C) = \frac{3}{2}$, $a = \sqrt{3}$, and $b+c = 3$, find the area of $\triangle ABC$.

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