
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
Given a positive integer  $ k$ , there is a positive integer  $ n$  with the property that one can obtain the sum of the first  $ n$  positive integers by writing some  $ k$  digits to the right of  $ n$ . Find the remainder of  $ n$  when dividing at  $ 9$ .

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