
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 triangle $ABC,$ points $D,$ $E,$ and $F$ are on sides $\overline{BC},$ $\overline{CA},$ and $\overline{AB},$ respectively, with $BD:DC = 1:3,$ $CE:EA = 2:1,$ and $AF:FB = 1:2.$ Line segments $\overline{AD},$ $\overline{BE},$ and $\overline{CF}$ intersect at points $P,$ $Q,$ and $R,$ respectively. Compute the ratio of the area of triangle $PQR$ to the area of triangle $ABC.$

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