
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 \( XY \parallel BC \) and \( XY \) is tangent to \( \omega \). What is the largest possible integer length for \( XY \)?

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