
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 rectangle $ABCD,$ $AB=8$ and $BC=4$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$, are chosen such that $BE=2EF=2FC$. These segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at points $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s,$ and $t$ is $1$. What is $r+s+t$?
A) 52
B) 55
C) 58
D) 60

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