
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
Let a hyperbola and a parabola share a focus, with the directrix of the parabola being the line that contains the minor axis of the hyperbola. They intersect at two points. Given that the equation of the hyperbola is \(\frac{x^2}{16} - \frac{y^2}{9} = 1\), find the distance between the two intersection points.

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