
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 30-foot golden rope to a majestic castle's cylindrical turret whose radius is 10 feet. The rope is fixed to the turret at ground level and to the unicorn at a height of 5 feet. If the rope is taut and the end of the rope opposite to the turret is 5 feet from the nearest point on the turret, find the length of the rope that is in contact with the turret expressed as \(\frac{a - \sqrt{b}}{c}\) feet, where \(a\), \(b\), and \(c\) are positive integers with \(c\) being a prime number. Determine \(a+b+c\).

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