
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 Cartesian coordinate system $(xOy)$, the parametric equations of circle $C$ are given by $\begin{cases} x = -5 + \sqrt{2}\cos t \\ y = 3 + \sqrt{2}\sin t \end{cases}, (t \text{ is the parameter})$, and in the polar coordinate system established with the origin $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis, the polar equation of line $l$ is $\rho \cos (\theta + \frac{\pi}{4}) = -\sqrt{2}$. The polar coordinates of points $A$ and $B$ are $A(2, \frac{\pi}{2})$ and $B(2, \pi)$, respectively.

1. Find the standard equation of circle $C$ and the rectangular equation of line $l$.
2. Point $P$ is any point on circle $C$. Find the minimum area of $\triangle PAB$.

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