
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 circumscribed circle \(\omega\) of triangle \(ABC\). The line \(AC\) intersects the circumscribed circle 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 the segment \(AE\)?

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