
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 $\{a_n\}$ that satisfies $a_n + S_n = n$, and the sequence $\{b_n\}$ that satisfies $b_1 = a_1$ and $b_n = a_n - a_{n-1}$ for $n \geqslant 2$:

(1) Find the general term formula for $\{a_n\}$.

(2) Find the general term formula for $\{b_n\}$.

(3) Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $B_n$.

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