
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
Let $A$ be an $n$-digit integer written in the decimal system. Consider a family of numbers derived from $A$ as follows: For every natural number $k < n$, represent $A$ in the form:

$$
A=10^{k} B+C
$$

where $C < 10^{k}$ and $B, C$ are integers, and let the new number be

$$
A_{k}=10^{n-k} C+B
$$

Determine all natural numbers that have the property that if a six-digit number is divisible by them, then every member of its family is also divisible by them.

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