
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 and each consists of a minimum period of 30 digits. In the decimal representation of  $a-b$ , the period is at least  $15$ . Find the minimum value of  $k\in\mathbb{N}$  such that, in the decimal representation of  $a+kb$ , the length of period is at least  $15$ . 

*A. Golovanov*

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