
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 the right triangle \(ABC\), the acute angle at vertex \(A\) is \(\alpha\). A line is drawn through the midpoint \(D\) of the hypotenuse \(AB\) and intersects the leg \(AC\) at point \(E\). In what ratio does this line divide the area of triangle \(ABC\), given that \(\angle DEA = \beta\) and \(AE > 0.5 AC\)?

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