
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 Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\begin{cases} x=2+ \sqrt {7}\cos \alpha \\ y= \sqrt {7}\sin \alpha\end{cases}$ (with $\alpha$ as the parameter). In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the polar equation of the curve $C_{2}$ is $\rho=8\cos \theta$, and the polar equation of the line $l$ is $\theta= \frac {\pi}{3} (\rho\in\mathbb{R})$.
(Ⅰ) Find the polar equation of the curve $C_{1}$ and the Cartesian equation of the line $l$;
(Ⅱ) If the line $l$ intersects $C_{1}$ and $C_{2}$ at points $A$ and $B$ in the first quadrant respectively, and $P$ is a moving point on $C_{2}$, find the maximum area of $\triangle PAB$.

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