
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  $\mathcal S$  be a set of  $16$  points in the plane, no three collinear. Let  $\chi(S)$  denote the number of ways to draw  $8$  lines with endpoints in  $\mathcal S$ , such that no two drawn segments intersect, even at endpoints. Find the smallest possible value of  $\chi(\mathcal S)$  across all such  $\mathcal S$ . 

*Ankan Bhattacharya*

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