
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 $f(x)= \sqrt{3}\sin^2x + \sin x\cos x$.
$(1)$ Find the smallest positive period of the function $f(x)$;
$(2)$ Given that the interior angles $A$, $B$, and $C$ of $\triangle ABC$ have their respective opposite sides $a$, $b$, and $c$, if $f\left(\frac{A}{2}\right)= \frac{\sqrt{3}}{2}$, $a=4$, $b+c=5$, find the area of $\triangle ABC$.

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