
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  $ABC$  be a fixed triangle in the plane. Let  $D$  be an arbitrary point in the plane. The circle with center  $D$ , passing through  $A$ , meets  $AB$  and  $AC$  again at points  $A_b$  and  $A_c$  respectively. Points  $B_a, B_c, C_a$  and  $C_b$  are defined similarly. A point  $D$  is called good if the points  $A_b, A_c,B_a, B_c, C_a$ , and  $C_b$  are concyclic. For a given triangle  $ABC$ , how many good points can there be?

(A. Garkavyj, A. Sokolov ) 

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