
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
Selected Exercise $(4-4)$: Coordinate System and Parametric Equations

In the rectangular coordinate system $(xOy)$, the parametric equations of curve $(C_{1})$ are given by $\begin{cases} x = 2\cos\alpha \\ y = 2 + 2\sin\alpha \end{cases} (\alpha \text{ is the parameter})$, where $M$ is a moving point on $(C_{1})$, $P$ is a point such that $\overrightarrow{OP} = 2\overrightarrow{OM}$. The locus of point $P$ forms the curve $(C_{2})$.

(I) Find the equation of $(C_{2})$;

(II) In the polar coordinate system with $O$ as the pole and the positive half of the $x$-axis as the polar axis, let $A$ be the intersection point of ray $\theta = \frac{\pi}{3}$ with $(C_{1})$ (excluding the pole), and $B$ be the intersection point of ray $\theta = \frac{\pi}{3}$ with $(C_{2})$ (excluding the pole). Find $|AB|$.

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