
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  $n$  be a positive integer. Let  $S$  be a subset of points on the plane with these conditions:  $i)$   There does not exist  $n$  lines in the plane such that every element of  $S$  be on at least one of them.   $ii)$  for all  $X \in S$  there exists  $n$  lines in the plane such that every element of  $S - {X} $  be on at least one of them.

Find maximum of  $\mid S\mid$ .

*Proposed by Erfan Salavati*

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