
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
Consider a parabola and an ellipse that intersect at two points and share a focus. The directrix of the parabola is the horizontal line passing through the center of the ellipse, and it is perpendicular to the ellipse's major axis. The ellipse is given by the equation \(\frac{x^2}{16} + \frac{y^2}{36} = 1\). Find the distance between the two intersection points of the parabola and the ellipse.

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