
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
A set  $A_1 , A_2 , A_3 , A_4$  of 4 points in the plane is said to be *Athenian* set if there is a point  $P$  of the plane satsifying 

(*)  $P$  does not lie on any of the lines  $A_i A_j$  for  $1 \leq i < j \leq 4$ ;
(**) the line joining  $P$  to the mid-point of the line  $A_i A_j$  is perpendicular to the line joining  $P$  to the mid-point of  $A_k A_l$ ,  $i,j,k,l$  being distinct.

(a) Find all *Athenian* sets in the plane.
(b) For a given *Athenian* set, find the set of all points  $P$  in the plane satisfying (*) and (**)

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