
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 eccentricity of the ellipse $E: \frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1\left(a > b > 0\right)$ is $\frac{\sqrt{2}}{2}$, and point $A(1, \sqrt{2})$ is on this ellipse.
(1) Find the equation of ellipse $E$;
(2) If the line $l$ with slope $\sqrt{2}$ intersects ellipse $E$ at two distinct points $B$ and $C$, find the maximum area of $\triangle ABC$.

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