
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 $ABC$, point $E$ lies on $\overline{AC}$ such that $AE:EC = 1:3$, and point $F$ lies on $\overline{AB}$ such that $AF:FB = 1:4$. Let point $P$ be the intersection of $\overline{BE}$ and $\overline{CF}$. Find the coordinates of point $P$ as a barycentric coordinate relative to $A$, $B$, and $C$, where the coordinates satisfy $x+y+z=1$.

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