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

(1) Find the range of $f(x)$ when $x\in[0, \frac{\pi}{2} ]$.

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

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