
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 $D$ lies on $\overline{BC}$ extended past $C$ such that $BD:DC = 2:1$, and $E$ lies on $\overline{AC}$ such that $AE:EC = 2:1$. Let $P$ be the intersection of lines $BE$ and $AD$. Express $\overrightarrow{P}$ as a linear combination of $\overrightarrow{A}$, $\overrightarrow{B}$, and $\overrightarrow{C}$, where the weights sum to 1.

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