
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 that the coordinates of the two foci of an ellipse are $(-\sqrt{2}, 0)$ and $(\sqrt{2}, 0)$, and it passes through the point $(\frac{\sqrt{2}}{2}, \frac{\sqrt{30}}{6})$.
(1) Find the standard equation of the ellipse;
(2) If a line $l$ with slope $k$ passes through the point $(0, -2)$, and intersects the ellipse at two distinct points $A$ and $B$, find the maximum area of $\triangle OAB$.

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