
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 all integer numbers  $n\ge 4$  which satisfy the following condition: from every  $n$  different  $3$ -element subsets of  $n$ -element set it is possible to choose  $2$  subsets, which have exactly one element in common.

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