
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 $f(x) = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) - 2\sqrt{3}\sin^2\left(\frac{x}{2}\right) + \sqrt{3}$,  
(Ⅰ) The graph of $y=f(x)$ can be obtained by which transformations of the graph of $y=\sin x$?  
(Ⅱ) If the graph of $y=f(x+\varphi)$ has a center of symmetry at $\left(\frac{\pi}{3}, 0\right)$, what is the value of $\varphi$?  
(Ⅲ) Let $g(x) = f(x) + \sin x$ reach its maximum value when $x=\theta$. Find $\cos\theta$.

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