
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
On graph paper, a square measuring $11 \times 11$ cells is drawn. It is required to mark the centers of some cells in such a way that the center of any other cell lies on the segment connecting any two marked points that are on the same vertical or horizontal line. What is the minimum number of cells that need to be marked?

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