
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
In the plane, there are three circles \( C_{i} (i=1,2,3) \), where \( C_{1} \) has a diameter \( AB = 1 \); \( C_{2} \) shares the same center as \( C_{1} \) and has a diameter \( k \) such that \( 1 < k < 3 \); and \( C_{3} \) is centered at \( A \) with a diameter \( 2k \) (with \( k \) as a fixed value). Consider all line segments \( XY \) where one end \( X \) is on \( C_{2} \) and the other end \( Y \) is on \( C_{3} \), and the segment \( XY \) passes through point \( B \). For which ratio \( \frac{XB}{BY} \) is the length of the segment \( XY \) minimized?

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