
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 hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{20} = 1$, an ellipse $(C)$ has the foci of the hyperbola as its vertices and its vertices as the foci. The left and right vertices of the ellipse $(C)$ are $A$ and $B$, respectively, and $P(\frac{3}{2}, \frac{5\sqrt{3}}{2})$.
1. Find the equation of ellipse $(C)$.
2. Let point $M$ be on the major axis $AB$ of the ellipse. The distance from point $M$ to line $AP$ is equal to $|MB|$. Find the minimum value of the distance $d$ from points on the ellipse to point $M$.

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