
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 $x^{2}- \frac{y^{2}}{b^{2}}=1(b > 0)$ with left and right foci $F_{1}$ and $F_{2}$. A line $l$ passes through $F_{2}$ and intersects the hyperbola at points $A$ and $B$.
1. If line $l$ has an angle of inclination of $\frac{\pi}{2}$ and $\triangle F_{1}AB$ is an equilateral triangle, find the equations of the asymptotes of the hyperbola.
2. If $b=\sqrt{3}$ and the slope of line $l$ exists such that $(\overrightarrow{F_{1}A}+ \overrightarrow{F_{1}B})\cdot \overrightarrow{AB}=0$, find the slope of line $l$.

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