
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 parabola $y^{2}=4\sqrt{3}x$ whose directrix intersects with the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ ($a > 0, b > 0$) at points $A$ and $B$, and one of the asymptotes of the hyperbola is $y=\sqrt{2}x$. If point $F$ is the focus of the parabola, and $\triangle FAB$ is an equilateral triangle, then the standard equation of the hyperbola is \_\_\_\_\_\_.

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