
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
Three circles  $C_i$  are given in the plane:  $C_1$  has diameter  $AB$  of length  $1$ ;  $C_2$  is concentric and has diameter  $k$  ( $1 < k < 3$ );  $C_3$  has center  $A$  and diameter  $2k$ .  We regard  $k$  as fixed.  Now consider all straight line segments  $XY$  which have one endpoint  $X$  on  $C_2$ , one endpoint  $Y$  on  $C_3$ , and contain the point  $B$ .  For what ratio  $XB/BY$  will the segment  $XY$  have minimal length?

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