
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 Zeta follows an elliptical orbit with its sun at one focus. At its closest approach (perigee), it is 3 astronomical units (AU) from the sun, and at its farthest point (apogee), it is 15 AU away. Calculate the distance from the sun to Zeta when Zeta is exactly halfway along its orbit from perigee to apogee.

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