
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
Given a semicircle with diameter \(AB = 2R\) and center \(O\), two parallel lines are drawn through points \(A\) and \(O\). These lines intersect the circumference of the semicircle at points \(C\) and \(D\) respectively, and form an angle \(x\) with the diameter \(AB\). Draw the chords \(BC, BD\), and \(CD\), creating the trapezoid \(AODC\) and the triangle \(BCD\). Determine the angle \(x\) such that:

1. The sum of the areas of the trapezoid and the triangle is \(S^2\). What is the maximum value of \(S^2\)?
2. The area of the trapezoid is twice the area of the triangle.
3. The area of the triangle is twice the area of the trapezoid.

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