
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
Let \( P \) be a point inside triangle \( \triangle ABC \). Perpendiculars are drawn from point \( P \) to the sides \( BC \), \( CA \), and \( AB \), intersecting at points \( D \), \( E \), and \( F \) respectively. Find all points \( P \) such that \( \frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF} \) is minimized.

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