
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 $AB$ such that the ratio AP:PB = 2:1. Let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. Calculate the ratio of the area of triangle $BPD$ to that of triangle $ABC$, denoted by $r$.
A) $\frac{1}{12}$
B) $\frac{1}{18}$
C) $\frac{1}{24}$
D) $\frac{1}{36}$

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