
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 on the surface of a sphere with center $O$ and radius $15$. The lengths of $AB=14$, $BC=15$, and $CA=13$. Determine the distance from $O$ to the plane of $\triangle ABC$, represented 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. What is $p+q+r$?

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