
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
In the rectangular coordinate system $(xOy)$, curve $(C_1)$ is defined by the parametric equations: $\begin{cases} x = 2 + \sqrt{7} \cos \alpha \\ y = \sqrt{7} \sin \alpha \end{cases}$ where $\alpha$ is a parameter. Curve $(C_2)$ has the polar coordinate equation $\rho = 8 \cos \theta$, and line $(l)$ has the polar coordinate equation $\theta = \frac{\pi}{3}$.

(I) Find the polar coordinate equation of curve $(C_1)$ and the rectangular coordinate equation of line $(l)$.

(II) Suppose line $(l)$ intersects curves $(C_1)$ and $(C_2)$ at points $A$ and $B$ in the first quadrant, respectively. Let $P$ be a moving point on $(C_2)$. Find the maximum area of $\Delta PAB$.

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