
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  $ n$  be a positive integer and  $ [ \ n ] = a.$  Find the largest integer  $ n$  such that the following two conditions are satisfied:

 $ (1)$   $ n$  is not a perfect square;

 $ (2)$   $ a^{3}$  divides  $ n^{2}$ .

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