
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  $|S|\ge 2$  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  $n\ge 2$ , 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. 

*Proposed by Carl Schildkraut*

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