
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 a trapezoid \(ABCD\), where \(BC\) and \(AD\) are the bases, the diagonal \(AC\) is the angle bisector of \(\angle BAD\), which is equal to \(120^\circ\). The circumradius of triangle \(ABD\) is \(\sqrt{3}\). Diagonals \(AC\) and \(BD\) intersect at point \(O\). The areas of triangles \(AOD\) and \(BOC\) are in the ratio 4:1. Find all sides of the trapezoid \(ABCD\).

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