
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
Some vertices (the vertices of the unit squares) of a \(6 \times 6\) grid are colored red. We need to ensure that for any sub-grid \(k \times k\) where \(1 \leq k \leq 6\), at least one red point exists on its boundary. Find the minimum number of red points needed to satisfy this condition.

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