
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's call integer square-free if it's not divisible by  $p^2$  for any prime  $p$ . You are given a square-free integer  $n>1$ , which has exactly  $d$  positive divisors. Find the largest number of its divisors that you can choose, such that  $a^2 + ab - n$  isn't a square of an integer for any  $a, b$  among chosen divisors.

*(Proposed by Oleksii Masalitin)*

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