
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
The quadrilateral $ABCD$ is inscribed in a circle $\omega$, whose center lies on the side $AB$. The circle $\omega_{1}$ is externally tangent to the circle $\omega$ at point $C$. The circle $\omega_{2}$ is tangent to the circles $\omega$ and $\omega_{1}$ at points $D$ and $E$ respectively. The line $BC$ intersects the circle $\omega_{1}$ again at point $P$, and the line $AD$ intersects the circle $\omega_{2}$ again at point $Q$. It is known that points $P, Q,$ and $E$ are distinct. Find the angle $\angle PEQ$.

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