
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 a positive integer that, in decimal representation, has digits only 0 or 1, we call this number a "good number". Determine the smallest positive integer \( k \) such that for any positive integer \( n \), it can be expressed as \( n = a_{1} \pm a_{2} \pm \ldots \pm a_{k} \), where \( a_{i} (1 \leq i \leq k) \) are good numbers.

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