
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
Let an ellipse E be defined by the equation $$\frac {y^{2}}{a^{2}} + \frac {x^{2}}{b^{2}} = 1 \quad (a > b > 0)$$, with eccentricity $e = \frac{\sqrt{2}}{2}$, and let point M $\left(\frac{\sqrt{2}}{2}, -1\right)$ lie on the ellipse.

(1) Find the equation of the ellipse E.
(2) A line passing through point M $(-2, 0)$ intersects the ellipse E at points A and B. Let O be the origin. Find the maximum area of triangle $\triangle AOB$ and the equation of the line when the maximum area is achieved.

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