
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 a triangle with  $AB=13$ ,  $BC=14$ , and  $AC=15$ .  Denote by  $\omega$  its incircle.  A line  $\ell$  tangent to  $\omega$  intersects  $\overline{AB}$  and  $\overline{AC}$  at  $X$  and  $Y$  respectively.  Suppose  $XY=5$ .  Compute the positive difference between the lengths of  $\overline{AX}$  and  $\overline{AY}$ .

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