
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$ ,  $ k$ ,  $ p$  be positive integers with  $ 2 \le k \le \frac {n}{p \plus{} 1}$ . Let  $ n$  distinct points on a circle be given. These points are colored blue and red so that exactly  $ k$  points are blue and, on each arc determined by two consecutive blue points in clockwise direction, there are at least  $ p$  red points. How many such colorings are there?

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