
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
A semi-circle with center $O$ and radius $r$ has $AB$ as its diameter. A rectangle $BCDE$ is inscribed in the semi-circle such that $E$ and $D$ lie on the circle and $BC = DE = r/2$. Let $S$ represent the area of the circular sector formed by $OED$ and let $T$ represent the area of triangle $OED$. Calculate the ratio $T:S$ as $BC = DE$ is half of $AB$.
A) $\sqrt{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{\sqrt{2}}{\pi}$
D) $\frac{1}{2\pi}$
E) $1$

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