
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
The left and right foci of the hyperbola $x^2 - \frac{y^2}{b^2} = 1$ ($b > 0$) are $F_1$ and $F_2$, respectively. A line $l$ passes through $F_2$ and intersects the hyperbola at points $A$ and $B$.
(1) If the slope angle of $l$ is $\frac{\pi}{2}$ and $\triangle F_1AB$ is an equilateral triangle, find the equation of the asymptotes of the hyperbola;
(2) Let $b = \sqrt{3}$. If the slope of $l$ exists, $M$ is the midpoint of $AB$, and $\overrightarrow{FM} \cdot \overrightarrow{AB} = 0$, find the slope of $l$.

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