
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xoy$, the parametric equation of circle $C$ is $\begin{cases} & x=-5+\sqrt{2}\cos t \\ & y=3+\sqrt{2}\sin t \end{cases}$, ($t$ is the parameter), in the polar coordinate system with the origin $O$ as the pole and the non-negative half-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 $A,B$ are respectively $A(2,\frac{\pi }{2}),B(2,\pi)$

(1) Find the standard equation of circle $C$ and the Cartesian equation of line $l$;

(2) Point $P$ is any point on circle $C$, find the minimum value of the area of $\Delta PAB$.

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