
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 a positive integer  $n$ ,  $\sigma(n)$  denotes the sum of the positive divisors of  $n$ . Determine   $$ \limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}} $$  **Note:** Given a sequence ( $a_n$ ) of real numbers, we say that  $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$  if ( $a_n$ ) is not upper bounded, and, otherwise,  $\limsup\limits_{n\rightarrow \infty} a_n$  is the smallest constant  $C$  such that, for every real  $K > C$ , there is a positive integer  $N$  with  $a_n < K$  for every  $n > N$ .

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