
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  $n \ge 3$  be an integer and  $S$  be a set of  $n$  elements. Determine the largest integer  $k_n$  such that: for each selection of  $k_n$   $3-$ subsets of  $S$ , there exists a way to color elements of  $S$  with two colors such that none of the chosen  $3-$ subset is monochromatic.

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