
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 a hyperbola with the equation $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, its eccentricity is $2$, and the distance from the coordinate origin to line $AB$ is $\frac{3}{2}$, where $A(a, 0)$ and $B(0, -b)$.

1. Find the equation of the hyperbola.
2. If $B_1$ is the endpoint of the conjugate axis of the hyperbola on the positive half of the $y$-axis, and a line is drawn through $B$ intersecting the hyperbola at points $M$ and $N$, find the equation of line $MN$ when $B_1M \perp B_1N$.

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