
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
An ellipse and a hyperbola share a focus, and the directrix of the parabola is the line $x = 5$ (a vertical line to the right of the center of the ellipse). The ellipse and the parabola intersect at two points. Given that the equation of the ellipse is \(\frac{x^2}{16} + \frac{y^2}{4} = 1\), find the distance between the points where the parabola and the ellipse intersect.

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