
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passing through point $P(\frac{\sqrt{2}}{2}, \frac{1}{2})$, with its left focus at $F_1(-\frac{\sqrt{2}}{2}, 0)$.

(I) Find the equation of the ellipse $C$.

(II) Line $y = x + m (m \neq 0)$ intersects ellipse $C$ at two distinct points $A$ and $B$. With the origin $O$, find the maximum area of $\triangle OAB$.

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