
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  $AB$  be a chord on parabola  $y=x^2$  and  $AB||Ox$ . For each point C on parabola different from  $A$  and  $B$  we are taking point  $C_1$  lying on the circumcircle of  $\triangle ABC$  such that  $CC_1||Oy$ . Find a locus of  points  $C_1$ .

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