
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
Triangle  $A_0B_0C_0$  is given in the plane. Consider all triangles  $ABC$  such that:
(i) The lines  $AB,BC,CA$  pass through  $C_0,A_0,B_0$ , respectvely,
(ii) The triangles  $ABC$  and  $A_0B_0C_0$  are similar.
Find the possible positions of the circumcenter of triangle  $ABC$ .


Austrian-Polish 2000

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