
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}
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