
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 $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ passes through points $M(2,0)$ and $N(0,1)$.

$(1)$ Find the equation of ellipse $C$ and its eccentricity;

$(2)$ A line $y=kx (k \in \mathbb{R}, k \neq 0)$ intersects ellipse $C$ at points $A$ and $B$, point $D$ is a moving point on ellipse $C$, and $|AD| = |BD|$. Does the area of $\triangle ABD$ have a minimum value? If it exists, find the equation of line $AB$; if not, explain why.

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