
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 plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\left\{{\begin{array}{l}{x=6-\frac{{\sqrt{3}}}{2}t}\\{y=\frac{1}{2}t}\end{array}}\right.$ (where $t$ is a parameter). Taking the coordinate origin as the pole and the non-negative $x$-axis as the polar axis to establish a polar coordinate system, the polar equation of the curve $C$ is $\rho =6\cos \theta $.  
$(1)$ Find the general equation of the line $l$ and the rectangular coordinate equation of the curve $C$;  
$(2)$ Suppose the line $l$ intersects the curve $C$ at points $A$ and $B$, and point $P$ is a moving point on the curve $C$, find the maximum area of $\triangle PAB$.

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