
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 \( n = 6k \). Construct a family of 3-element subsets \(\mathcal{A} = \{A_1, A_2, \ldots, A_l\}\) of \( X = \{1, 2, \ldots, n\} \) with \( l = \frac{n^2}{6} \) such that each 2-element subset of \( X \) is contained in at least one member of \(\mathcal{A}\).

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