
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
The planet Zorion travels in an elliptical orbit around its sun, which is positioned at one focus of the ellipse. Zorion's closest approach to its sun is 3 astronomical units (AU), and its furthest point is 15 AU away. If the entire orbit of Zorion is tilted by 30 degrees from its original plane, calculate the distance from Zorion to its sun when Zorion is exactly halfway through its orbit from perigee to apogee.

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