
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 $P$, $Q$, and $R$ lie on the surface of a sphere with center $S$ and radius $24$. It is given that $PQ=20$, $QR=21$, and $RP=29$, and that the distance from $S$ to $\triangle PQR$ 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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