
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
**Problem 3.** Let  $n\geq 3$  is given natural number, and  $M$  is the set of the first  $n$  primes. For any nonempty subset  $X$  of  $M$  with  $P(X)$  denote the product of its elements. Let  $N$  be a set of the kind  $\ds\frac{P(A)}{P(B)}$ ,  $A\subset M, B\subset M, A\cap B=\emptyset$  such that the product of any 7 elements of  $N$  is integer. What is the maximal number of elements of  $N$ ?

*Alexandar Ivanov*

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