
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 an equilateral triangle $\triangle DEF$. Points $G$, $H$, and $I$ are the feet of the perpendiculars from $P$ to $\overline{DE}$, $\overline{EF}$, and $\overline{FD}$, respectively. Given that $PG=2$, $PH=3$, and $PI=4$, what is $DE$ in terms of radicals?

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