
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
The points \(A\), \(B\), and \(C\) are positioned on the surface of a sphere with center \(O\) and radius \(30\). The side lengths of the triangular configuration defined by these points are \(AB = 30\), \(BC = 40\), and \(CA = 50\). It is also given that the distance from \(O\) to triangle \(ABC\) is expressible in the form \(\frac{p\sqrt{q}}{r}\), where \(p\), \(q\), and \(r\) are positive integers, \(p\) and \(r\) are relatively prime, and \(q\) is not divisible by the square of any prime. Determine \(p+q+r\).

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