
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
An ellipse $C:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{6}}{3}$. $(F_{1},F_{2})$ are the left and right foci of the ellipse, and $A$ is its lower vertex. The intersection points of the circle with center $(F_{1})$, radius $\sqrt{3}+1$, and the circle with center $(F_{2})$, radius $\sqrt{3}-1$ are on the ellipse $C$.

(1) Find the equation of the ellipse $C$;

(2) Determine whether there exists a real number $k$ such that the parallelogram formed by the sides $AM$ and $AN$ is a rhombus, where $M$ and $N$ are the two distinct intersection points of the line $l:y=kx+\frac{3}{2}$ and the ellipse $C$. If such $k$ exists, find its value; if not, explain the reason.

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