
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 \(C\) lie on the surface of a sphere with center \(O\) and radius \(15\). It is given that \(AB = 7\), \(BC = 24\), \(CA = 25\), and that the distance from \(O\) to \(\triangle ABC\) is \(\frac{m\sqrt{n}}{k}\), where \(m\), \(n\), and \(k\) are positive integers, \(m\) and \(k\) are relatively prime, and \(n\) is not divisible by the square of any prime. Find \(m+n+k\).

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