
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 $f(x)=4\sin x\cdot \cos^2\left( \frac{x}{2}+ \frac{\pi}{4}\right)-\cos 2x$.  
$(1)$ The graph of the function $y=f(2x)$ is translated to the right by $\frac{\pi}{6}$ units to obtain the graph of the function $y=g(x)$. Find the range of the function $g(x)$ when $x\in\left[ \frac{\pi}{12}, \frac{\pi}{2}\right]$.  
$(2)$ Given that $a$, $b$, $c$ are respectively the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, and it satisfies $b=2$, $f(A)= \sqrt{2}-1$, $\sqrt{3}a=2b\sin A$,  
$B\in(0, \frac{\pi}{2})$, find the area of $\triangle ABC$.

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