
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 $F_1$ and $F_2$ are the left and right foci, respectively, of an ellipse $E$ with an eccentricity of $\frac{\sqrt{2}}{2}$. The ellipse is defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b > 0$. A chord of length $\sqrt{2}$ is intercepted by the ellipse on a line passing through $F_2$ and perpendicular to the $x$-axis.

1. Find the equation of the ellipse $E$.
2. Let the line $l: y = x + m$ intersect with the ellipse $E$ at points $A$ and $B$. The perpendicular bisector of segment $AB$ intersects the $x$-axis at point $P$. Find the maximum area of $\triangle PAB$ as $m$ varies.

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