
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
We call a permutation  $ \left(a_1, a_2, ..., a_n\right)$  of  $ \left(1, 2, ..., n\right)$  *quadratic* if there exists at least a perfect square among the numbers  $ a_1$ ,  $ a_1 \plus{} a_2$ ,  $ ...$ ,  $ a_1 \plus{} a_2 \plus{} ... \plus{} a_n$ . Find all natural numbers  $ n$  such that all permutations in  $ S_n$  are quadratic.

*Remark.*  $ S_{n}$  denotes the  $ n$ -th symmetric group, the group of permutations on  $ n$  elements.

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