
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 three diameters of a sphere that are mutually perpendicular. Divide each diameter into 4 equal parts. Through each division point, draw a plane perpendicular to the corresponding diameter. How many regions does the sphere’s surface get divided into by the 9 circles formed by these 9 planes?

- How many regions does the sphere’s surface get divided into if each of the 3 diameters is divided into 6 equal parts, and we proceed similarly?

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