
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 M, whose axes of symmetry align with the coordinate axes, and a parabola with equation $x^{2}=-4\sqrt{2}y$, share a focus with the ellipse M. Additionally, given that point A $(1, \sqrt{2})$ lies on ellipse M,  
(I) find the equation of ellipse M;  
(II) given a line l with direction vector $(1, \sqrt{2})$, which intersects ellipse M at points B and C, find the maximum value of the area of triangle ABC.

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