
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
A circle passes through vertices \( A \) and \( B \) of an acute-angled triangle \( ABC \), intersecting side \( AC \) at point \( X \) and side \( BC \) at point \( Y \). It is given that this circle passes through the circumcenter of triangle \( XCY \). Segments \( AY \) and \( BX \) intersect at point \( P \). It is known that \( \angle ACB = 2 \angle APX \). Find the angle \( ACB \).

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