
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
Given a fixed triangle \( A B C \). Let \( D \) be an arbitrary point in the plane of the triangle that does not coincide with its vertices. A circle with center at \( D \) passing through \( A \) intersects lines \( A B \) and \( A C \) at points \( A_{b} \) and \( A_{c} \) respectively. Similarly defined are points \( B_{a}, B_{c}, C_{a} \), and \( C_{b} \). A point \( D \) is called good if the points \( A_{b}, A_{c}, B_{a}, B_{c}, C_{a} \), and \( C_{b} \) lie on a single circle.

How many points can be good for a given triangle \( A B C \)?

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