
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
You are given a square  $n \times n$ . The centers of some of some  $m$  of its  $1\times 1$  cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer  $n \geq 3$ , find the largest value of  $m$  for which it is possible.

*Proposed by Oleksiy Masalitin, Fedir Yudin*

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