
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 20.  It is given that  $AB=13, BC=14, CA=15,$  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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