
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
Let \( A = \{1, 2, \cdots, n\} \), and let \( S_{n} \) denote the sum of all elements in the non-empty proper subsets of \( A \). Let \( B_{n} \) represent the number of subsets of \( A \). Find the value of \( \lim_{n \rightarrow +\infty} \frac{S_{n}}{n^{2} \cdot B_{n}} \).

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