
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 extension of side $BC$ of triangle $ABC$, point $D$ is taken such that line $AD$ is tangent to the circumscribed circle $\omega$ of triangle $ABC$. Line $AC$ intersects the circumscribed circle of triangle $ABD$ at point $E$, with $AC:CE = 1:2$. It turns out that the angle bisector of angle $ADE$ is tangent to circle $\omega$. Find the angles of triangle $ABC$.

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