
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
There are  $ k$  rooks on a  $ 10 \times 10$  chessboard. We mark all the squares that at least one rook can capture (we consider the square where the rook stands as captured by the rook). What is the maximum value of  $ k$  so that the following holds for some arrangement of  $ k$  rooks: after removing any rook from the chessboard, there is at least one marked square not captured by any of the remaining rooks.

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