
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
Consider the hyperbola given by the equation $x^{2}- \frac{y^{2}}{b^{2}}=1 (b > 0)$, with left and right foci $F_{1}$ and $F_{2}$, respectively. A line $l$ passes through $F_{2}$ and intersects the hyperbola at points $A$ and $B$.

1. If the angle of inclination of line $l$ is $\frac{\pi}{2}$ and $\triangle F_{1}AB$ is an equilateral triangle, find the equations of the asymptotes of the hyperbola.
2. Let $b = \sqrt{3}$. If the slope of line $l$ exists, $M$ is the midpoint of $AB$, and $\overrightarrow{FM} \cdot \overrightarrow{AB} = 0$, find the slope of line $l$.

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