
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 \( n \) and \( k \) be positive integers such that \( n > k \geq 2 \). Consider coloring each of the \( n \) points on a circle either red or blue such that any \( k \) consecutive points contain at least one red point. Let \( f_k(n) \) be the number of such valid coloring schemes. Determine the necessary and sufficient conditions on \( n \) and \( k \) such that \( f_k(n) \) is an even number.

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