
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  $S$  be the set of all positive integers  $n$  such that each of the numbers  $n + 1$ ,  $n + 3$ ,  $n + 4$ ,  $n + 5$ ,  $n + 6$ , and  $n + 8$  is composite. Determine the largest integer  $k$  with the following property: For each  $n \in S$  there exist at least  $k$  consecutive composite integers in the set
{ $n, n +1, n + 2, n + 3, n + 4, n + 5, n + 6, n + 7, n + 8, n + 9$ }.

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