
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 expressed as decimal fractions, each with a minimal period consisting of 30 digits. The decimal representation of the number \( a - b \) has a minimal period length of 15. For what smallest natural \( k \) can the minimal period length of the decimal representation of the number \( a + k b \) also be equal to 15?

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