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

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ When the area of $\triangle AMN$ is $\frac{4\sqrt{2}}{5}$, find the value of $k$.

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