
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with one focus at $F(1, 0)$ and an eccentricity of $\frac{\sqrt{2}}{2}$. Let $P$ be a moving point on the major axis of the ellipse $C$, and $l$ be a line passing through point $P$ with a slope of $1$, which intersects the ellipse at points $A$ and $B$.

1. Find the equation of the ellipse $C$.
2. Find the maximum value of $|PA|^2 + |PB|^2$.

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