
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
Denote  $\mathbb{Z}_{>0}=\{1,2,3,...\}$  the set of all positive integers. Determine all functions  $f:\mathbb{Z}_{>0}\rightarrow \mathbb{Z}_{>0}$  such that, for each positive integer  $n$ , $\hspace{1cm}i) \sum_{k=1}^{n}f(k)$  is a perfect square, and $\vspace{0.1cm}$  $\hspace{1cm}ii) f(n)$  divides  $n^3$ .

*Proposed by Dorlir Ahmeti, Albania*

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