
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
1) Cells of  $8 \times 8$  table contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer  $a$  from the table there exists integer written in the same row or in the same column such that it is not relatively prime with  $a$ . Find maximum possible number of prime integers in the table.

2) Cells of  $2n \times 2n$  table,  $n \ge 2,$  contain pairwise distinct positive integers. Each integer is prime or a product of two primes. It is known that for any integer  $a$  from the table there exist integers written in the same row and in the same column such that they are not relatively prime with  $a$ . Find maximum possible number of prime integers in the table.

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