
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 parallel to the major axis of the ellipse. The parabola and ellipse intersect at two points. Given that the equation of the ellipse is $\frac{x^2}{36} + \frac{y^2}{16} = 1$, find the distance between these two intersections.

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