
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 $ABC$, let $D$ lie on $\overline{BC}$ such that $BD:DC = 2:3$, and let $E$ lie on $\overline{AB}$ such that $AE:EB = 4:1$. Let $P$ be the intersection of lines $BD$ and $AE$.

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