
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, $F(\sqrt{2}, 0)$ is its right focus, and the line perpendicular to the $x$-axis passing through $F$ intersects the ellipse at a chord of length $2$.
$(1)$ Find the equation of the ellipse $C$;
$(2)$ The line $l: y = kx + m (km \neq 0)$ intersects the ellipse $C$ at points $A$ and $B$. If the midpoint of segment $AB$ lies on the line $x + 2y = 0$, find the maximum area of $\triangle FAB$.

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