
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
Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $M$ and $B$, and let a line through $P$ intersect $BC$ at a point $D$. If the ratio of the area of triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, find $r$.
A) $0 < r < \frac{1}{4}$
B) $0 < r < \frac{1}{2}$
C) $\frac{1}{4} < r < \frac{1}{2}$
D) $\frac{1}{2} \leq r < 1$
E) $r = \frac{1}{2}$

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