
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 $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, where the left vertex is $A_{1}$, and the right focus is $F_{2}$. A line perpendicular to the $x$-axis passing through point $F_{2}$ intersects the ellipse at points $M$ and $N$. The slope of line $A_{1}M$ is $\dfrac {1}{2}$.  
$(1)$ Find the eccentricity of the ellipse;  
$(2)$ If the circumcircle of $\triangle A_{1}MN$ touches the ellipse at another point $D$ besides $M$, and the area of $\triangle F_{2}MD$ is $\dfrac {12}{7}$, find the equation of the ellipse.

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