
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
Given the cube Q, the union of the faces S, and the union of distinct planes p1, p2,...,pk intersecting the interior of Q, find the difference between the maximum and minimum possible values of k.

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