
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
As \( n \rightarrow +\infty \), find the limits of the following functions:
1) \( S_{1}(n)=\frac{1}{n}+\frac{2}{n}+\frac{3}{n}+\ldots+\frac{n-1}{n} \);
2) \( S_{2}(n)=\frac{1}{n^{2}}+\frac{2}{n^{2}}+\frac{3}{n^{2}}+\ldots+\frac{n-1}{n^{2}} \);
3) \( S_{3}(n)=\frac{1}{n^{3}}+\frac{2}{n^{3}}+\frac{3}{n^{3}}+\ldots+\frac{n-1}{n^{3}} \).

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