
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
Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $E$ lies on $\overline{AB}$ so that $EB=2$, point $G$ lies on $\overline{BC}$ so that $CG=2$, and point $F$ lies on $\overline{CD}$ so that $DF=1$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?
A) $\frac{1}{9\sqrt{2}}$
B) $\frac{1}{18}$
C) $\frac{1}{10\sqrt{2}}$
D) $\frac{1}{6}$

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