
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 $ABC,$ $D$ is placed on $\overline{BC}$ extended past $C$ such that $BD:DC = 4:1,$ and $E$ is on $\overline{AC}$ such that $AE:EC = 2:1.$ Determine the point $P$, the intersection of lines $BE$ and $AD,$ in the coordinates of $A,$ $B,$ and $C$ using the vector representation
\[\overrightarrow{P} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C},\]
where $x,$ $y,$ and $z$ are constants such that $x + y + z = 1.$ Provide the ordered triple $(x,y,z).$

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