
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 vertices at $(0,1)$, and its eccentricity is $\dfrac{\sqrt{3}}{2}$.

$(1)$ Find the equation of ellipse $C$;  
$(2)$ From a point $P$ on the circle $x^{2}+y^{2}=16$, draw two tangents to the ellipse $C$, with the points of tangency being $A$ and $B$. When the line $AB$ intersects the $x$-axis and $y$-axis at points $M$ and $N$ respectively, find the minimum value of $|MN|$.

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