
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 $E: \dfrac{y^{2}}{12} + \dfrac{x^{2}}{4} = 1$, two different lines passing through the point $P(0,-2)$ intersect the ellipse $E$ at points $A$, $B$, and $C$, $D$, where $A$ is the right vertex of the ellipse.
1. Find the equation of the circle with diameter $AB$.
2. Suppose the circles with diameters $AB$ and $CD$ intersect at points $M$ and $N$. Investigate whether the line $MN$ passes through a fixed point. If it does, find the coordinates of the fixed point. If it does not, explain why.

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