
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 \((x, y)\), the parametric equation of circle \(C\) is
$$
\begin{cases}
x=-5+ \sqrt {2}\cos t\\
y=3+ \sqrt {2}\sin t
\end{cases},
$$
where \(t\) is the parameter. 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 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 Cartesian equation of line \(l\).  
(2) For any point \(P\) 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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