
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 provided diagram, the center of the circle is \(O\), and the radius is \(2r\) feet. Chord \(PQ\) is parallel to chord \(RS\) such that \(O\), \(M\), \(N\), \(L\) are collinear, and \(M\) is the midpoint of \(PQ\). Let \(T_1\) (sq. ft.) represent the area of trapezoid \(RSQP\) and let \(U\) (sq. ft.) represent the area of rectangle \(UVWP\). As \(PQ\) and \(RS\) are moved so that \(OM\) increases toward the value \(2r\), while maintaining \(NL = LM\), the ratio \(T_1:U\) becomes arbitrarily close to:
A) \( \sqrt{3} \)
B) \( \frac{1}{2}+\sqrt{3} \)
C) \( \frac{1}{2}+\frac{\sqrt{3}}{2} \)
D) \( 1+\sqrt{3} \)
E) \( \frac{1}{\sqrt{3}}+\frac{1}{2} \)

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