
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 $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$, one of its vertices is $A(2,0)$, and its eccentricity is $\dfrac{\sqrt{2}}{2}$. 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 $\Delta AMN$ is $\dfrac{4\sqrt{2}}{5}$, find the value of $k$.

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