
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 \(ABCD\), the base \(AD\) is equal to 16, the sum of the side \(AB\) and the diagonal \(BD\) is equal to 40, and the angle \(CBD\) is \(60^\circ\). The ratio of the areas of triangles \(ABO\) and \(BOC\), where \(O\) is the intersection point of the diagonals, is 2. Find the area of the trapezoid.

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