
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
For  $n$  positive integers  $a_1,...,a_n$  consider all their pairwise products  $a_ia_j$ ,  $1 \le i < j \le n$ . Let  $N$  be the number of those products which are the cubes of positive integers.
Find the maximal possible value of  $N$  if it is known that none of  $a_j$  is a cube of an integer.

(S. Mazanik)

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