
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 number and let \( A \) be a set of positive integers that satisfies the following conditions:
(i) the set of prime divisors of the elements in \( A \) consists of \( p-1 \) elements;
(ii) for any nonempty subset of \( A \), the product of its elements is not a perfect \( p \)-th power.
What is the largest possible number of elements in \( A \)?

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