
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
Initially, there is a natural number \( N \) written on the board. At any moment, Misha can choose a number \( a > 1 \) on the board, erase it, and write down all of its natural divisors except for \( a \) itself (the same numbers can appear multiple times on the board). After some time, it turned out that there were \( N^2 \) numbers written on the board. For which \( N \) could this happen?

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