
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
Points \(A, B, C\), and \(D\) lie on a line \(\ell\) in that order, with \(AB = CD = 4\) and \(BC = 8\). Circles \(\Omega_1\), \(\Omega_2\), and \(\Omega_3\) with diameters \(AB\), \(BC\), and \(CD\), respectively, are drawn. A line through \(A\) and tangent to \(\Omega_3\) intersects \(\Omega_2\) at the two points \(X\) and \(Y\). Find the length of \(XY\).

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