
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 $f(x)=\cos x\sin \left(x-\dfrac{\pi }{6}\right)$.  
(I) When $x\in[0,\dfrac{\pi}{2}]$, find the range of the function $f(x)$.  
(II) In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$, if $f(A)=\dfrac{1}{4}$, $a=\sqrt{3}$, and $\sin B=2\sin C$, find the area of $\triangle ABC$.

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