
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
An ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(2,1)$, with an eccentricity of $\frac{\sqrt{2}}{2}$.

(1) Find the equation of the ellipse $E$;

(2) Line $l: y = x + m$ intersects $E$ at points $A$ and $B$. Determine if there exists a point $C$ on the $y$-axis such that $\triangle ABC$ is a right triangle. If it exists, find the equation of line $l$; otherwise, explain the reason.

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