
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  $S$  be a finite set of points in the plane. A linear partition of  $S$  is an unordered pair  $\{A,B\}$  of subsets  of  $S$  such that  $A\cup B=S,\ A\cap B=\emptyset,$  and  $A$  and  $B$  lie on opposite sides of some straight line disjoint from  $S$  ( $A$  or  $B$  may be empty). Let  $L_{S}$  be the number of linear partitions of  $S.$  For each positive integer  $n,$  find the maximum of  $L_{S}$  over all sets  $S$  of  $n$  points.

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