
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 vectors $\overrightarrow{m}=(\sqrt{3}\sin x, 1-\sqrt{2}\sin x)$ and $\overrightarrow{n}=(2\cos x, 1+\sqrt{2}\sin x)$.
$(1)$ If the function $f(x)=\overrightarrow{m}\cdot\overrightarrow{n}$, when $x\in[0, \frac{\pi}{2}]$, find the range of $f(x)$;
$(2)$ If in $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and it satisfies $\frac{b}{a}=\sqrt{3}$, $\frac{\sin B\cos A}{\sin A}=2-\cos B$, find the value of $f(B)$.

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