
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 parametric equation of line $l$ as $\begin{cases} & x=4+t \\\\ & y=t \\end{cases}$ ($t$ is the parameter), establish a polar coordinate system with the coordinate origin as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. The polar coordinate equation of circle $C$ is $\rho =4\cos \theta$. Line $l$ intersects with circle $C$ at points $A$ and $B$.

(1) Find the rectangular coordinate equation of circle $C$ and the length of chord $AB$;

(2) A moving point $P$ is on circle $C$ (not coinciding with $A$ or $B$), find the maximum value of the area of $\triangle ABP$.

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