
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 \( n \) be a positive integer and let \( f_1(x), \ldots, f_n(x) \) be affine functions from \( \mathbb{R} \) to \( \mathbb{R} \) such that, among the \( n \) graphs of these functions, no two are parallel and no three are concurrent. Let \( S \) be the set of all convex functions \( g(x) \) from \( \mathbb{R} \) to \( \mathbb{R} \) such that for each \( x \in \mathbb{R} \), there exists \( i \) such that \( g(x) = f_i(x) \).

Determine the largest and smallest possible values of \( |S| \) in terms of \( n \).

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