
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 the top base radius \( r_1 = 5 \) cm, bottom base radius \( r_2 = 10 \) cm, and slant height \( A_1A_2 = 20 \) cm. A string is tied at the midpoint of \( A_1A_2 \) and wound around the lateral surface of the cone to a point on the circumference of the bottom base. Find:
(1) The minimum length of the string.
(2) The minimum distance from a point on the string to a point on the circumference of the top base when the string is at its minimum length.

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