
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
Let  $ p$  be a prime,  $ n$  a natural number, and  $ S$  a set of cardinality  $ p^n$  . Let  $ \textbf{P}$  be a family of partitions of  $ S$  into nonempty parts of sizes divisible by  $ p$  such that the intersection of any two parts that occur in any of the partitions has at most one element. How large can  $ |\textbf{P}|$  be?

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