
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
A finite set \( S \) of points in the coordinate plane is called overdetermined if \( |S| \geq 2 \) and there exists a nonzero polynomial \( P(t) \), with real coefficients and of degree at most \( |S| - 2 \), satisfying \( P(x) = y \) for every point \( (x, y) \in S \).

For each integer \( n \geq 2 \), find the largest integer \( k \) (in terms of \( n \) ) such that there exists a set of \( n \) distinct points that is not overdetermined, but has \( k \) overdetermined subsets.

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