
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
The area of triangle $ABC$ is 1. Points $D$ and $E$ lie on sides $AB$ and $AC$, respectively. Lines $BE$ and $CD$ intersect at point $P$. The area of quadrilateral $BCED$ is twice the area of triangle $PBC$. Find the maximum value of the area of triangle $PDE$.

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