
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  $ABC$  be an isosceles triangle with  $\angle A = 90^{\circ}$ . Points  $D$  and  $E$  are selected on sides  $AB$  and  $AC$ , and points  $X$  and  $Y$  are the feet of the altitudes from  $D$  and  $E$  to side  $BC$ .  Given that  $AD = 48\sqrt2$  and  $AE = 52\sqrt2$ , compute  $XY$ .

*Proposed by Evan Chen*

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