
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
Convex quadrilateral $ABCD$ has $AB = 10$ and $CD = 15$. The diagonals $AC$ and $BD$ intersect at $E$. Given that $AC = 18$, and the area of $\triangle AED$ is twice the area of $\triangle BEC$, find the length of $AE$.
A) $9 - 9\sqrt{2}$  
B) $18 - 18\sqrt{2}$  
C) $9(2 - \sqrt{2})$  
D) $9(2 + \sqrt{2})$  
E) $18(1 + \sqrt{2})$

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