
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  $n \geq 2$  be an integer. Find the smallest positive integer  $m$  for which the following holds: given  $n$  points in the plane, no three on a line, there are  $m$  lines such that no line passes through any of the given points, and
for all points  $X  \neq Y$  there is a line with respect to which  $X$  and  $Y$  lie on opposite sides

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