
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

Two circles touch each other externally, and each of them touches a larger circle internally. The radius of one is half the radius of the largest circle, and the other is one-third the radius of the largest circle. Find the ratio of the length of the common internal tangent segment to the smaller circles, enclosed within the largest circle, to its diameter.

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