
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
If $AB$ and $CD$ are perpendicular diameters of circle $Q$, and $P$ is a point in $\overline{AQ}$ such that $\measuredangle QPC = 45^\circ$, then the length of $PQ$ divided by the length of $AQ$ is?
A) $\frac{\sqrt{3}}{2}$
B) $\frac{\sqrt{3}}{3}$
C) $\frac{\sqrt{2}}{2}$
D) $\frac{1}{2}$
E) $\frac{2}{3}$

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