
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
Given that $AB$ and $CD$ are perpendicular diameters of circle $Q$, with $P$ on $\overline{AQ}$ further extended beyond $Q$, and $\measuredangle CPQ = 45^\circ$, find the length of $PQ$ divided by the length of $AQ$.
A) $\sqrt{2}$
B) $\frac{\sqrt{2}}{2}$
C) $\frac{1}{\sqrt{2}}$
D) $\frac{\sqrt{3}}{3}$
E) $\frac{1}{2}$

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