
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-angled triangle \(ABC\), the acute angle \(A\) is equal to \(\alpha\) radians. An arc of a circle with its center at the vertex of the right angle \(C\) touches the hypotenuse at point \(D\) and intersects the legs \(AC\) and \(BC\) at points \(E\) and \(F\) respectively. Find the ratio of the areas of the curvilinear triangles \(ADE\) and \(BDF\).

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