
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 $ABC,$ $E$ lies on $\overline{AC}$ such that $AE:EC = 3:1,$ and $F$ lies on $\overline{AB}$ such that $AF:FB = 1:4.$ Let $P$ be the intersection of $\overline{BE}$ and $\overline{CF}.$ Determine the vector $\overrightarrow{P}$ written as $x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},$ where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$

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