
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
Points \(D\) and \(E\) lie on sides \(AB\) and \(AC\) of \(\triangle ABC\) with area 1, respectively. Line segments \(BE\) and \(CD\) intersect at point \(P\). Points \(D\) and \(E\) move on \(AB\) and \(AC\) respectively, but satisfy the condition that the area of quadrilateral \(BCED\) is twice the area of \(\triangle PBC\). Find the maximum area of \(\triangle PDE\).

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