
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 equations of the asymptotes of the hyperbola E: $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, where $a>0$ and $b>0$, are to be determined given that the distance between the intersections of a line perpendicular to the x-axis and the hyperbola E and the intersections of the line with the asymptotes of the hyperbola E is $\frac{\sqrt{3}}{2}$ times the distance between the intersections of the line with the asymptotes of the hyperbola E.

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