
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
1. (1) Given the parabola \( y = x^2 - a \) and the hyperbola \( y = \frac{1}{x} \) have three distinct intersection points, find the range of the real number \( a \).  
(2) In part (1), given all possible values of \( a \), determine the locus of the circumcenter of the triangle formed by the three intersection points.

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