
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
In a circle, $AB$ is a diameter, and $CD$ is a chord parallel to $AB$. Point $E$ is the intersection of lines $AC$ and $BD$, and $\angle AED = \beta$, but now $\beta$ is such that $\angle AED + \angle BED = 180^\circ - 2\beta$. The ratio of the area of triangle $\triangle CDE$ to that of triangle $\triangle ABE$ is required.

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