
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
Let \( k \) be a positive integer. Keira and Roland play a game of reverse chess. Initially, Roland chooses a positive integer \( n > \frac{k}{2023} \). Keira places \( k \) kings on \( k \) distinct squares of a \( 2023 \times n \) chess board. Then Roland places a rook on an unoccupied square of the board. Both players then take turns moving any number (possibly zero) of their pieces, with Keira starting first. Each king cannot move to a square occupied by another king, but it can capture the rook. Furthermore, if Keira chooses to move more than one king in her turn, she moves them one at a time. Roland's rook is not permitted to capture any king, nor may it pass through a square occupied by a king.

For which \( k \) can Keira guarantee to capture Roland's rook, regardless of Roland's moves or choice of \( n \)? 

(A king can move exactly one square in any horizontal, vertical or diagonal direction. A rook can move any number of squares in a horizontal or vertical direction. One piece (rook or king) captures another by moving to the square it occupies.)

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