
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
Circles of radius $4$ and $8$ are externally tangent to each other, and both are internally tangent to a circle of radius $12$. There exists a chord in the circle of radius $12$ that also serves as a common external tangent to the smaller two circles. Determine the square of the length of this chord.

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