
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  $A,B,C$  be points in that order along a line, such that  $AB=20$  and  $BC=18$ . Let  $\omega$  be a circle of nonzero radius centered at  $B$ , and let  $\ell_1$  and  $\ell_2$  be tangents to  $\omega$  through  $A$  and  $C$ , respectively. Let  $K$  be the intersection of  $\ell_1$  and  $\ell_2$ . Let  $X$  lie on segment  $\overline{KA}$  and  $Y$  lie on segment  $\overline{KC}$  such that  $XY\|BC$  and  $XY$  is tangent to  $\omega$ . What is the largest possible integer length for  $XY$ ?

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