
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
Positive rational numbers \(a\) and \(b\) are written as decimal fractions, each with a minimum period consisting of 30 digits. The decimal representation of the number \(a - b\) has a minimum period of 15. For what smallest natural number \(k\) can the minimum period of the decimal representation of the number \(a + kb\) also be equal to 15?

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