
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 positive integers \( n \) with the following property: the \( k \) positive divisors of \( n \) have a permutation \(\left( d_{1}, d_{2}, \ldots, d_{k} \right)\) such that for every \( i = 1, 2, \ldots, k \), the number \( d_{1} + \cdots + d_{i} \) is a perfect square.

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