
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
An integer  $n$  is said to be *good* if  $|n|$  is not the square of an integer. Determine all integers  $m$  with the following property:  $m$  can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.

*Proposed by Hojoo Lee, Korea*

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