
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 circle with center $O$ and radius $b$ inches, two parallel chords $CD$ and $EF$ are drawn such that $O$, $G$, $H$, $J$ are collinear. Chords are translated vertically such that $OG$ increases toward $b$ while keeping $JH=HG=1/2$ of the vertical distance between $CD$ and $EF$. The area of trapezoid $CDFE$ is denoted by $T$ and the area of rectangle $ELMF$ by $R$. Determine the ratio $T:R$ as $CD$ and $EF$ are translated upward such that the vertical distance between $CD$ and $EF$ doubles.
A) $\sqrt{2}$  
B) $1$  
C) $0$  
D) $\frac{1}{2}+\frac{\sqrt{2}}{2}$  
E) $\frac{1}{\sqrt{2}}+1$

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