
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 parabola \( y^2 = 2px \) with \( p > 0 \), two moving points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) are on the parabola, and \( O \) is the origin. Additionally, \( OA \perp OB \).

1. Find the equation of the trajectory \( C \) of the midpoint of segment \( AB \).
2. If the distance from a point on \( C \) to the line \( x - 2y + 2 \sqrt{5} - p = 0 \) is \( d \), find the minimum value of \( d \).

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