
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 $17$. It is given that $PQ=12$, $QR=16$, $RP=20$, and that the distance from $S$ to $\triangle PQR$ is $\frac{x\sqrt{y}}{z}$, where $x$, $y$, and $z$ are positive integers, $x$ and $z$ are relatively prime, and $y$ is not divisible by the square of any prime. Find $x+y+z$.

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