
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 an ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\ (a > b > 0)$, its two foci and one endpoint of the minor axis form an equilateral triangle. If the maximum distance from a focus to a point on the ellipse is $3\sqrt{3}$, find the standard equation of the hyperbola with real semi-axis $a$ and imaginary semi-axis $b$ where the foci lie on the $y$-axis.

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