
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
Suppose  $ABCD$  is a trapezoid with  $AB\parallel CD$  and  $AB\perp BC$ . Let  $X$  be a point on segment  $\overline{AD}$  such that  $AD$  bisects  $\angle BXC$  externally, and denote  $Y$  as the intersection of  $AC$  and  $BD$ . If  $AB=10$  and  $CD=15$ , compute the maximum possible value of  $XY$ .

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