
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 $$C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$$ shares a focus with the parabola $$y^2 = 4\sqrt{2}x$$. The area of the quadrilateral formed by connecting the four vertices of the ellipse is $$2\sqrt{3}$$.
(1) Find the equation of ellipse C;
(2) The line $l: y = kx + m (k \neq 0)$ intersects ellipse C at two distinct points M and N. Let A be the endpoint of the short axis of ellipse C on the negative y-axis. If triangle AMN is an isosceles triangle with MN as its base, find the range of values for m.

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