
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  $A$  and  $B$  be points with  $AB=12$ . A point  $P$  in the plane of  $A$  and  $B$  is  $\textit{special}$  if there exist points  $X, Y$  such that


- $P$  lies on segment  $XY$ ,
- $PX : PY = 4 : 7$ , and
- the circumcircles of  $AXY$  and  $BXY$  are both tangent to line  $AB$ .

A point  $P$  that is not special is called  $\textit{boring}$ .
Compute the smallest integer  $n$  such that any two boring points have distance less than  $\sqrt{n/10}$  from each other.

*Proposed by Michael Ren*

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