
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
The secant passing through the intersection points of two circles, one with center \( O_1 \) and radius \( 4 \text{ cm} \), and the other with center \( O_2 \) and radius \( 6 \text{ cm} \), intersects the segment \( O_1O_2 \) at point \( T \). It is known that the length of \( O_1O_2 \) is not less than \( 6 \text{ cm} \). The larger circle intersects the segment \( O_1O_2 \) at point \( A \), while the smaller circle intersects it at point \( B \), with the ratio \( AT : BT = 1 : 2 \). Calculate the length of the segment \( O_1O_2 \).

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