
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
Given a right triangle $ABC$. On the extension of the hypotenuse $BC$, a point $D$ is chosen such that the line $AD$ is tangent to the circumcircle $\omega$ of triangle $ABC$. The line $AC$ intersects the circumcircle of triangle $ABD$ at point $E$. It turns out that the angle bisector of $\angle ADE$ is tangent to the circle $\omega$. In what ratio does point $C$ divide segment $AE$?

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