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

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