
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 $25$-foot golden rope to the base of a magician'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 $5$ feet. The unicorn has pulled the rope taut, the end of the rope is $5$ feet from the nearest point on the tower, and the length of the rope that is touching the tower is $\frac{d-\sqrt{e}}f$ feet, where $d, e,$ and $f$ are positive integers, and $f$ is prime. Find $d+e+f.$

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