
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 the set \( A = \{ 1, 2, \cdots, n \} \). Let \( S_n \) denote the sum of all elements in the non-empty proper subsets of \( A \), and \( B_n \) denote the number of subsets of \( A \). Find the limit \(\lim_{{n \to \infty}} \frac{S_n}{n^2 B_n}\).

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