
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: $$\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1 (a > b > 0)$$, with F$(\sqrt {2}, 0)$ as its right focus, and the chord formed by the intersection of the line perpendicular to the x-axis through F and the ellipse has a length of 2.  
(1) Find the equation of ellipse C;  
(2) The line $l: y = kx + m (km \neq 0)$ intersects 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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