
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
A circle with center \(O\) has radius 20. Chords \( \overline{AB} \) of length 24 and \( \overline{CD} \) of length 16 intersect at point \( P \). The distance between the midpoints of the two chords is 10. Find \( \overline{OP}^2 \), which can be expressed as \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers.

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