
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
For any natural number  $m$ , we denote by  $\phi (m)$  the number of integers  $k$  relatively prime to  $m$  and satisfying  $1 \le k \le m$ . Determine all positive integers  $n$  such that for every integer  $k > n^2$ , we have  $n | \phi  (nk + 1)$ .

(Daniel Harrer)

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