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

(Ⅰ) Find the equation of the ellipse $C$;  
(Ⅱ) When the area of $\triangle AMN$ is $\dfrac{\sqrt{10}}{3}$, find the value of $k$.

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