
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 an ellipse $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\ (a > b > 0)$ with one of its vertices at $A(2,0)$, an eccentricity of $\frac{\sqrt{2}}{2}$, and a line $y=k(x-1)$ intersecting the ellipse $C$ at two distinct points $M$ and $N$.

1. Find the equation of the ellipse $C$.
2. Find the value of the real number $k$ when the area of $\triangle AMN$ is $\frac{\sqrt{10}}{3}$.

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