
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
You are given two diameters \( A B \) and \( C D \) of circle \( \Omega \) with radius 1. A circle is drawn in one of the smaller sectors formed such that it is tangent to \( A B \) at \( E \), tangent to \( C D \) at \( F \), and tangent to \( \Omega \) at \( P \). Lines \( P E \) and \( P F \) intersect \( \Omega \) again at \( X \) and \( Y \). What is the length of \( X Y \), given that \( A C = \frac{2}{3} \)?

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