
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 \( ABC \) (\( AB = BC \)). Point \( E \) is marked on ray \( BA \) beyond point \( A \), and point \( D \) is marked on 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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