
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
A parabola and an ellipse share a focus, and the directrix of the parabola is the line containing the minor axis of the ellipse. The parabola and ellipse intersect at four points. Given that the equation of the ellipse is $\frac{x^2}{36} + \frac{y^2}{16} = 1$, find the sum of the distances between the pairs of intersection points on opposite sides of the ellipse.

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