
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 trapezoid \(ABCD\), side \(AD\) is the longer base. It is known that \(AD = CD = 4 \frac{2}{3}\), \(\angle BAD = 90^\circ\), and \(\angle BCD = 150^\circ\). On base \(AD\), triangle \(AED\) is constructed such that points \(B\) and \(E\) are on the same side of line \(AD\) and \(AE = DE\). The height of this triangle, drawn from vertex \(E\), is \(1 \frac{2}{5}\). Find the area of the common part of trapezoid \(ABCD\) and triangle \(AED\).

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