
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
In some cells of a $10 \times 10$ board, $k$ rooks are placed, and then all the cells that are attacked by at least one rook (the rook also attacks the cell it stands on) are marked. What is the maximum value of $k$ for which, after removing any rook from the board, there will be at least one marked cell that is no longer under attack?

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