
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 hyperbola \( x^2 - \frac{y^2}{4} = 1 \):
1. Two perpendicular rays are drawn from the center \( O \) and intersect the hyperbola at points \( A \) and \( B \). Find the equation of the locus of the midpoint \( P \) of segment \( AB \).
2. Find the distance from the center \( O \) of the hyperbola to the line \( AB \).

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