
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 $AB = 8$ and $BC = 6$. Point $E$ lies on $\overline{AB}$ such that $EB = 3$, point $G$ lies on $\overline{BC}$ such that $CG = 2$, and point $F$ lies on $\overline{CD}$ such that $DF = 4$. Segments $\overline{AG}$ and $\overline{AC}$ intersect $\overline{EF}$ at points $Q$ and $P$, respectively. What is the value of $\frac{PQ}{EF}$?
A) $\frac{1}{10\sqrt{37}}$
B) $\frac{1}{35\sqrt{37}}$
C) $\frac{5}{45\sqrt{37}}$
D) $\frac{4}{45\sqrt{37}}$
E) $\frac{2}{35\sqrt{37}}$

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