
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
Let  $\Omega$  and  $\omega$  be circles with radii  $123$  and  $61$ , respectively, such that the center of  $\Omega$  lies on  $\omega$ . A chord of  $\Omega$  is cut by  $\omega$  into three segments, whose lengths are in the ratio  $1 : 2 : 3$  in that order. Given that this chord is not a diameter of  $\Omega$ , compute the length of this chord.

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