
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
Points $A$ and $C$ lie on a circle centered at $P$, which is inside $\triangle ABC$ such that $\overline{AP}$ is perpendicular to $\overline{BC}$ and $\triangle ABC$ is equilateral. The circle intersects $\overline{BP}$ at $D$. If $\angle BAP = 45^\circ$, what is $\frac{BD}{BP}$?
A) $\frac{2}{3}$
B) $\frac{2 - \sqrt{2}}{2}$
C) $\frac{1 - \sqrt{2}}{2}$
D) $\frac{\sqrt{2}}{2}$

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