
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 $f(x)=2\cos x\cdot \sin \left( x-\dfrac{\pi }{6} \right)-\dfrac{1}{2}$.
(Ⅰ) Find the minimum value and the smallest positive period of the function $f(x)$;
(Ⅱ) Let $\Delta ABC$ be a triangle with sides $a, b, c$ opposite to angles $A, B, C$ respectively, and $c=\sqrt{3}$. If angle $C$ satisfies $f(C)=0$ and $\sin B=2\sin A$, find the values of $a$ and $b$.

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