
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
Fix an integer  $n\geq4$ . Let  $C_n$  be the collection of all  $n$ –point configurations in the plane, every three points of which span a triangle of area strictly greater than  $1.$  For each configuration  $C\in C_n$  let  $f(n,C)$  be the maximal size of a subconfiguration of  $C$  subject to the condition that every pair of distinct points has distance strictly greater than  $2.$  Determine the minimum value  $f(n)$  which  $f(n,C)$  achieves as  $C$  runs through  $C_n.$ *Radu Bumbăcea and Călin Popescu*

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