
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
Given the sequence \(\{a_n\}\):
\[1,1,2,1,2,3,\cdots, 1,2,\cdots, n,\cdots\]

Let \(S_n\) denote the sum of the first \(n\) terms of the sequence \(\{a_n\}\). Find all positive real number pairs \((\alpha, \beta)\) such that
\[
\lim_{n \rightarrow +\infty} \frac{S_n}{n^\alpha} = \beta.
\]

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