
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
Find all functions  $f:\mathbb{Z} \rightarrow \mathbb{Z}$  such that  $f(1) \neq f(-1)$  and   $$ f(m+n)^2 \mid f(m)-f(n) $$   for all integers  $m, n$ .

*Proposed by Liam Baker, South Africa*

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