
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
What is the smallest number \( k \) such that \( k \) cells of an \( 8 \times 9 \) board can be marked in such a way that no matter how a four-cell shape is placed on the board (it can be rotated and flipped), it will always cover at least one marked cell?

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