
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
Given a natural number \( N \geq 3 \), we define a set of \( N \) points on the coordinate plane as permissible if their x-coordinates are distinct and each of these points is colored either red or blue. We say that a polynomial \( P(x) \) separates a permissible set of points if either there are no red points above the graph of \( P(x) \) and no blue points below it, or vice versa (points of both colors may lie on the graph itself). What is the smallest \( k \) such that any permissible set of \( N \) points can be separated by a polynomial of degree at most \( k \)?

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