
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 right trapezium $ABCD$ with right angles at $A$ and $B$, $AB=6$ (the shorter base), and $CD=9$ (the longer base), the height from $A$ and $B$ to $CD$ is $4$. Points $E$ and $F$ divide the vertical segment $BD$, with $BE:EF:FD = 2:3:5$. Line segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BC}$ at points $P$ and $Q$ respectively. Determine the area ratio $\frac{\text{area of triangle } APB}{\text{area of triangle } BPC}$.
A) $\frac{1}{2}$
B) $1$
C) $2$
D) $\frac{3}{2}$

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