
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
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{6}}{3}$, and it passes through point $M(-3,-1)$.

(I) Find the equation of the ellipse $C$;

(II) If line $l:x-y-2=0$ intersects with ellipse $C$ at points $A$ and $B$, and $P$ is a moving point on ellipse $C$, find the coordinates of point $P$ when the area of $\triangle PAB$ is maximum, and find the maximum area of $\triangle PAB$.

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