
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, where $(A\_1, A\_2)$ are the left and right vertices of the hyperbola, and $M(X\_0, Y\_0)$ is a point on the hyperbola other than the two vertices. The product of the slopes of the lines $MA\_1$ and $MA\_2$ is $\frac{144}{25}$. Find the eccentricity of the hyperbola. If the distance from the focus of the hyperbola to the asymptote is $12$, find the equation of the hyperbola.

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