
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 integer  $k$  ( $k \ge 2$ ), for which there exists an integer  $n$  ( $n \ge k$ ) such that from any collection of  $n$  consecutive positive integers one can always choose  $k$  numbers, which verify the following conditions:

1. each chosen number is not divisible by  $6$ , by  $7$ , nor by  $8$ ;
2. the positive difference of any two distinct chosen numbers is not divisible by at least one of the
numbers  $6$ ,  $7$ , and  $8$ .

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