
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
For a positive integer  $n$  and a nonzero digit  $d$ , let  $f(n, d)$  be the smallest positive integer  $k$ , such that  $kn$  starts with  $d$ . What is the maximal value of  $f(n, d)$ , over all positive integers  $n$  and nonzero digits  $d$ ? 

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