
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 quadrilateral $ABCD$ satisfies $\angle ABC = 120^\circ, \angle ACD = 90^\circ, AC = 10\sqrt{2},$ and $CD = 20.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE = 5\sqrt{2}.$ Calculate the area of quadrilateral $ABCD$.

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