
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ with eccentricity $e = \frac{2\sqrt{3}}{3}$, the line $l$ passes through points $A(a, 0)$ and $B(0, -b)$, and the distance from the origin $O$ to line $l$ is $\frac{\sqrt{3}}{2}$.

1. Find the equation of the hyperbola.
2. Draw a line $m$ through point $B$ intersecting the hyperbola at points $M$ and $N$. If $\overrightarrow{OM} \cdot \overrightarrow{ON} = -23$, find the equation of line $m$.

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