
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 centered at $O$, point $A$ is on the circle, and $\overline{BA}$ is tangent to the circle at $A$. Triangle $ABC$ is right-angled at $A$ with $\angle ABC = 45^\circ$. The circle intersects $\overline{BO}$ at $D$. Chord $\overline{BC}$ also extends to meet the circle at another point, $E$. What is the value of $\frac{BD}{BO}$?
A) $\frac{3 - \sqrt{3}}{2}$
B) $\frac{\sqrt{3}}{2}$
C) $\frac{2 - \sqrt{2}}{2}$
D) $\frac{1}{2}$
E) $\frac{\sqrt{2}}{2}$

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