
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 two sequences of sets $\{A_{n}\}$ and $\{B_{n}\}$ defined as follows:

- $A_{1} = \varnothing$
- $B_{1} = \{0\}$
- $A_{n+1} = \{x + 1 \mid x \in B_{n}\}$
- $B_{n+1} = (A_{n} \cup B_{n}) \backslash (A_{n} \cap B_{n})$

Find all positive integers $n$ such that $B_{n} = \{0\}$.

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