
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 center of hyperbola G is at the origin of the coordinate system, its major axis lies on the x-axis, and its eccentricity is $\frac{\sqrt{5}}{2}$, and the difference in the distances from a point on G to the two foci of G is 12, determine the equation of the hyperbola G.

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