
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 \(24\)-foot golden rope to the base of a wizard's cylindrical tower whose radius is \(10\) feet. The rope is attached to the tower at ground level and to the unicorn at a height of \(6\) feet. The unicorn has pulled the rope taut, the end of the rope is \(6\) feet from the nearest point on the tower, and the unicorn is restrained by a spell to walk only along a semi-circular path around the tower. Find the length of the rope that is touching the tower in the form \(\frac{p-\sqrt{q}}{r}\) feet, where \(p, q,\) and \(r\) are positive integers, and \(r\) is prime. Then, calculate \(p+q+r\).

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