
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 rectangular coordinate system $xOy$, the equation of curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of $x$ as the polar axis. The polar equation of line $l$ is given by $\rho \sin(\theta - \frac{\pi}{4}) = -\sqrt{2}$.

1. Find the parametric equation of curve $C$ and the rectangular equation of line $l$.
2. If line $l$ intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively, and $P$ is a moving point on curve $C$, find the maximum area of $\triangle PAB$.

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