
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
For any point $P$ inside triangle $\triangle ABC$, let $D$, $E$, and $F$ be the intersections of lines $AP$, $BP$, and $CP$ with sides $BC$, $CA$, and $AB$ respectively. Find the point $P$ such that the area of $\triangle DEF$ is maximized.

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