
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 dragon is tethered by a 25-foot golden rope to the base of a sorcerer's cylindrical tower whose radius is 10 feet. The rope is attached to the tower at ground level and to the dragon at a height of 7 feet. The dragon 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}
user