
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
On the leg \( AC \) of the right triangle \( ABC \) with hypotenuse \( AB \), a point \( P \) is marked. Point \( D \) is the foot of the perpendicular dropped from vertex \( A \) to line \( BP \), and point \( E \) is the foot of the perpendicular dropped from point \( P \) to side \( AB \). On the plane, a point \( T \) is chosen such that lines \( TA \) and \( TP \) are tangents to the circumcircle of triangle \( PAB \). The point \( O \) is the center of this circumcircle. The perpendicular dropped from point \( T \) to line \( DE \) intersects line \( BC \) at point \( Q \), and the line passing through point \( C \) parallel to \( OQ \) intersects segment \( BO \) at point \( K \). Find the angle \( \angle OKE \).

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