
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  $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$  that satisfy the following two conditions:
[list] $\bullet\ f(n)$  is a perfect square for all  $n\in\mathbb{Z}_{>0}$  $\bullet\ f(m+n)=f(m)+f(n)+2mn$  for all  $m,n\in\mathbb{Z}_{>0}$ .[/list]

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