
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 the circle \( C: x^{2} + y^{2} = r^{2} \), the points \( P \) and \( P' \) are on the ray starting from the origin \( O \), and they satisfy \( |OP| |OP'| = r^{2} \). Find the equation satisfied by the symmetric point \( P' \) of a point \( P(x, y) \) on the hyperbola \( x^{2} - y^{2} = 1 \) with respect to the unit circle \( C: x^{2} + y^{2} = 1 \).

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