
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 and let  $G$  be the set of points  $(x, y)$  in the plane such that  $x$  and  $y$  are integers with  $1 \leq x, y \leq n$ . A subset of  $G$  is called *parallelogram-free* if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of  $G$  can have?

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