
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 $30$. It is given that $PQ=20$, $QR=21$, and $PR=29$, and that the distance from $S$ to $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime. Find $a+b+c$.

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