
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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