
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 the diameter of a circle and \( C \) be the center. Extend chord \( AX \) to \( Y \) such that \( AX = XY \) and denote the intersection of lines \( CY \) and \( BX \) as \( Z \). What is the locus of point \( Z \) if the line \( AX \) rotates around point \( A \)?

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