
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 $f(x)=\sqrt{3}\sin^{2}x+\sin x\cos x$.

(1) Find the range of $f(x)$ when $x\in \left[ 0,\frac{\pi }{3} \right]$.

(2) In triangle $\Delta ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $f(\frac{A}{2})=\frac{\sqrt{3}}{2}$, $a=4$, and $b+c=5$, find the area of $\Delta ABC$.

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