
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 $\begin{cases} x=4+ \frac { \sqrt {2}}{2}t \\ y= \frac { \sqrt {2}}{2}t\end{cases}$ (where $t$ is the parameter), with the origin as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of circle $C$ is $\rho=4\cos \theta$. Line $l$ intersects circle $C$ at points $A$ and $B$.  
$(1)$ Find the Cartesian 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 $B$), try to find the maximum area of $\triangle ABP$.

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