
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 any interior point of an equilateral triangle \( ABC \). Let \( D, E, \) and \( F \) be the feet of the perpendiculars dropped from \( P \) to the sides \( BC, CA, \) and \( AB \) respectively. Determine the value of the ratio

$$
\frac{PD + PE + PF}{BD + CE + AF}.
$$

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