
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
Suppose a hyperbola \( C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \) has a right focal point \( F \). Let \( P \) be a point outside the hyperbola. Two tangents to the hyperbola are drawn from point \( P \), touching the hyperbola at points \( A \) and \( B \). If \( A B \perp P F \), find the locus of point \( P \).

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