
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
A unicorn is tethered by a $15$-foot golden rope to the base of a wizard's cylindrical silo whose radius is $5$ feet. The rope is attached to the silo at ground level and to the unicorn at a height of $3$ feet. The unicorn has pulled the rope taut, the end of the rope is $3$ feet from the nearest point on the silo, and the length of the rope that is touching the silo is $\frac{p-\sqrt{q}}{r}$ feet, where $p, q,$ and $r$ are positive integers, and $r$ is prime. Find $p+q+r.$

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