
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
Find the largest  $n$  for which there exists a sequence  $(a_0, a_1, \ldots, a_n)$  of non-zero digits such that, for each  $k$ ,  $1 \le k \le n$ , the  $k$ -digit number  $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$  divides the  $(k+1)$ -digit number  $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$ .

P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.

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