
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 \( n \geq 4 \) and \( k, d \geq 2 \) be natural numbers with \( k \cdot d \leq n \). The \( n \) participants in the Mathematical Olympiad are sitting around a round table. Patrick wants to select \( k \) participants and send everyone else out of the room so that there are at least \( d-1 \) empty seats between every two of the remaining \( k \) participants. How many ways can Patrick do this, assuming all seats were initially occupied?

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