
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 an isosceles triangle \(ABC\) with \(AB = BC\). Point \(E\) is marked on the ray \(BA\) beyond point \(A\), and point \(D\) is marked on the side \(BC\). It is known that

\[
\angle ADC = \angle AEC = 60^\circ, \quad AD = CE = 13.
\]

Find the length of segment \(AE\) if \(DC = 9\).

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