
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
Let  $AB$  be a diameter of a circle and let  $C$  be a point on  $AB$  with  $2\cdot AC=BC$ . Let  $D$  and  $E$  be points on the circle such that  $DC\perp AB$  and  $DE$  is a second diameter. What is the ratio of the area of  $\triangle{DCE}$  to the area of  $\triangle{ABD}$ ?

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