
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 attacked by at least one rook are marked (it is assumed that a rook attacks the cell it is on). What is the largest \( k \) such that after removing any rook from the board, at least one marked cell will no longer be under attack?

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