
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 circle $Q$ with center $O$, diameters $AB$ and $CD$ are perpendicular. If point $P$ lies on diameter $CD$ with $\measuredangle APB = 45^\circ$, calculate the ratio of the length of $AP$ to the length of $AB$.
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{\sqrt{2}}{2}$
E) $\frac{2}{3}$

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