
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
Given a truncated cone with an upper base radius of $r_{1} = 5 \text{ cm}$, a lower base radius of $r_{2} = 10 \text{ cm}$, and a slant height $A_{1} A_{2} = 20 \text{ cm}$, a rope is pulled from the midpoint of $A_{1} A_{2}$ around the side surface of the truncated cone to a point on the circumference of the lower base at $A_{2}$. Find:
(1) The minimum length of the rope.
(2) When the rope is at its minimum length, the minimum distance from any point on the rope to a point on the circumference of the upper base.

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