
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
A coordinate system and parametric equations (4-4): In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are $\begin{cases} x=\frac{1}{2}+\frac{\sqrt{2}}{2}t, \\ y=2+\frac{\sqrt{2}}{2}t \end{cases}$ ($t$ is the parameter). In the polar coordinate system with the origin as the pole and the positive semi-axis of $x$ as the polar axis, the polar equation of curve $C$ is $\rho \sin ^{2} \theta - 2 \sin \theta - \rho = 0$.

(I) Find the general equation of line $l$ and the rectangular coordinate equation of curve $C$;

(II) Suppose line $l$ intersects curve $C$ at points $A$ and $B$. When point $P$ moves on the part of curve $C$ below line segment $AB$, find the maximum area of $\triangle PAB$.

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