
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  $ S$  be a set of  $ 100$  points in the plane.  The distance between every pair of points in  $ S$  is different, with the largest distance being  $ 30$ .  Let  $ A$  be one of the points in  $ S$ , let  $ B$  be the point in  $ S$  farthest from  $ A$ , and let  $ C$  be the point in  $ S$  farthest from  $ B$ .  Let  $ d$  be the distance between  $ B$  and  $ C$  rounded to the nearest integer.  What is the smallest possible value of  $ d$ ?

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