
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 $f(x)= \frac{\sqrt{3}}{2}\sin 2x - \cos^2 x - \frac{1}{2}, x \in \mathbb{R}.$
(Ⅰ) Find the minimum value of the function $f(x)$ and its smallest positive period.
(Ⅱ) Given $\triangle ABC$ with the sides opposite to angles A, B, and C being a, b, and c, respectively, and with $c= \sqrt{3}$ and $f(C) = 0.$ If $b = 2a,$ find the values of a and b.

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