
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 equilateral triangle \(ABC\), a circle is drawn passing through the center of the triangle and touching side \(BC\) at its midpoint \(D\). A line is drawn from point \(A\) tangent to the circle at point \(E\), such that \(\angle BAE < 30^{\circ}\). Find the area of triangle \(ABE\) if the area of triangle \(ABC\) is \(\frac{10}{4-\sqrt{2}}\).

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