
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 $ABC$, $D$ lies on $\overline{BC}$ extended past $C$ such that $BD:DC = 2:1$, and $E$ lies on $\overline{AC}$ such that $AE:EC = 3:2$. Let $P$ be the intersection of lines $BE$ and $AD$. Determine the vector $\overrightarrow{P}$ as a linear combination of $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{C}$, where $\overrightarrow{P} = x \overrightarrow{A} + y \overrightarrow{B} + z \overrightarrow{C}$ with $x + y + z = 1$. Enter the ordered triple $(x, y, z)$.

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