
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
A five-digit positive integer is called *$k$ -phobic* if no matter how one chooses to alter at most four of the digits, the resulting number (after disregarding any leading zeroes) will not be a multiple of  $k$ . Find the smallest positive integer value of  $k$  such that there exists a  $k$ -phobic number.

*Proposed by Yannick Yao*

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