
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 the quadrilateral pyramid \( P-ABCD \), it is given that \( AB \parallel CD \), \( AB \perp AD \), \( AB = 4 \), \( AD = 2\sqrt{2} \), \( CD = 2 \), and \( PA \perp \) plane \( ABCD \) with \( PA = 4 \). Let \( Q \) be a point on the segment \( PB \), and the sine of the angle between the line \( QC \) and the plane \( PAC \) is \( \frac{\sqrt{3}}{3} \). Find the value of \( \frac{PQ}{PB} \).

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