
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
Find the point \( P \) inside the triangle \( ABC \) for which
\[
\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}
\]
is minimal, where \( PD, PE, PF \) are the perpendiculars from \( P \) to \( BC, CA, \) and \( AB \) respectively.

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