
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
207. What is the smallest perimeter of a convex 32-sided polygon whose vertices all lie on the nodes of a grid with cell side length 1?

208*. a) In a $7 \times 7$ grid, you need to mark the centers of $k$ cells so that no four marked points form the vertices of a rectangle with sides parallel to the sides of the square. What is the largest possible $k$ for which this can be done?

b) Solve a similar problem for a $13 \times 13$ grid.

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