
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 that the two asymptotes of the hyperbola $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$ (where $a \gt 0$ and $b \gt 0$) intersect the directrices of the parabola $y^{2}=2px$ (where $p \gt 0$) at points $O$, $A$, and $B$, with $O$ being the origin, the eccentricity of the hyperbola is $2$, and the area of triangle $AOB$ is $\sqrt{3}$, determine the value of $p$.

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