
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 $\overrightarrow{a} = \left(\frac{1}{2}, \frac{1}{2}\sin x + \frac{\sqrt{3}}{2}\cos x\right)$ and $\overrightarrow{b} = (1, y)$, and knowing that $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$. Let the function $y=f(x)$.

(1) Find the analytical expression of the function $y=f(x)$.
(2) Suppose in an acute-angled triangle $\triangle ABC$, $f\left(A - \frac{\pi}{3}\right) = \sqrt{3}$ and side $BC = \sqrt{3}$, find the maximum perimeter of $\triangle ABC$.

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