
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
We say that some positive integer  $m$  covers the number  $1998$ , if  $1,9,9,8$  appear in this order as digits of  $m$ . (For instance  $1998$  is covered by  $2\textbf{1}59\textbf{9}36\textbf{98}$  but not by  $213326798$ .) Let  $k(n)$  be the number of positive integers that cover  $1998$  and have exactly  $n$  digits ( $n\ge 5$ ), all different from  $0$ . What is the remainder of  $k(n)$  on division by  $8$ ?

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