
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 equation of the circle $C$ is
$$
\begin{cases}
x=-5+ \sqrt{2}\cos t \\
y=3+ \sqrt{2}\sin t
\end{cases}
$$
where $t$ is the parameter, and in the polar coordinate system with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar equation of the line $l$ is $\rho \cos \left(\theta+ \dfrac{\pi}{4}\right)=-\sqrt{2}$. The polar coordinates of points $A$ and $B$ are respectively $A(2, \dfrac{\pi}{2})$ and $B(2, \pi)$.

(1) Find the general equation of circle $C$ and the Cartesian equation of the line $l$;
(2) Let point $P$ be any point on circle $C$, find the minimum value of the area of $\triangle PAB$.

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