
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
Point \( P \) is inside equilateral \( \triangle DEF \). Points \( X \), \( Y \), and \( Z \) are the feet of the perpendiculars from \( P \) to \( \overline{DE} \), \( \overline{EF} \), and \( \overline{FD} \), respectively. Given that \( PX = 2 \), \( PY = 3 \), and \( PZ = 4 \), what is \( DE \) in terms of radicals?

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